The problem with intuition is that it is notoriously unreliable. Humans suffer from an effect called 'observational bias', because one tends to remember the exceptional better than the common. This is probably the reason that GMs/world champions have grossly overestimated the tactical value of a King (as ~4 Pawns): in games where the King plays an important role it can indeed be very strong, but there are plenty of cases where a King is of no use at all (because it cannot catch up with a passed Pawn). These tend to be dismissed, as "the King played no role here, so we could not see how stong it really is". While in fact you could see how weak it was by its lack of ability to play a role. In practice two non-royal Kings are conclusively defeated by the Bishop pair, (in combination with balanced other material, and in particular sufficiently many Pawns). In games between computer programs that most humans could not beat at all. Of course none of these GMs ever played such a game even once.
Other forms of intuition often result from application of simplistic logic, rather than observation. It is 'intuitively obvious' that a BN is worth several Pawns less than RN, as B is worth several Pawns less than R, and it is their only difference. Alas, it is not true. They are almost equivalent. It ignores the effect that some moves can cooperate better than others, and in games BN + Pawn would score convincingly better than RN (and on average even beat Q).
We should also keep in mind that piece values are just an approximation. It is not a law of nature that the strength of an army can be obtained by adding a value of individual pieces, and that the win probability can be calculated from the difference between the thus obtained army strength. And indeed, closer study shows that it is not true at all. The win probability depends on how well pieces in the army cooperate, and complement each other, and how effective they are against what the opponent has.
For example, A=BN and C=RN are more effective against a Queen than against a combination of lighter material (say R+N+2P) that in itself would perfectly balance a Queen. Because all squares attacked by the latter, even though very similar to the number of squares attacked by a single Q, are no-go areas for a C or A, even when they are protected, while they would not have to shy away from a Q attack in similar situations. This causes Q+C+A < R+B+C+A, in Capablanca Chess, even though Q > R+B as usual. The extra C and A on the Q side are effectively weaker pieces than their counterparts on the R+B side, so much that it reverses the advantage. An extreme manifestation of this effect is that 7 Knights easily beat 3 Queens on an 8x8 board. Something that cannot be explained by any value for N/Q that would make sense in a context with more mixed FIDE material.
Your claim that B+2P ~ R and N+2P < R, which I don't doubt, cannot be used to conclude that B > N because of these subtleties. Piece values are not defined as how well the pieces do against a Rook, but by how well they do against a mix of opponent pieces such as these typically occur in end-games. And I have no doubt that the average performance of the Bishop suffers from the fact that there are many cases where B+2P ~ B+P, while N+2P would have done much better (namely when the Bishops are on unlike shades).
Note that the claim lone B ~ N was not based on what I would call a 'computer study'. I have no doubt a computer was used in the process, but just as an aid for quickly searching a huge database of human GM games. Not by playing computers against each other. The fact that a computer was used thus in no way had any effect on the conclusion. In the Kaufman study the claim was detailed further by stating that the B-N difference correlated with the number of Pawns, and exact equality only occurred when each player had about 5 Pawns; for fewer Pawns the Bishop performed better, for more Pawns the Knight. It is also common knowledge that Knights typically perform poorer in end-games where there are Pawns on different wings than when all Pawns are close together. This is of course also something that transcends piece values, which are defined as the best estimate for the chances without knowing the location of the pieces. Piece values are not the only terms that contribute to the heuristic evaluation of individual positions.
But to come back to the main topic: I don't think it would be a good idea to dismis any form of a quality standard on published piece values because "people should know that they should not believe what they read". That is an argument that could be used for publishing any form of fake news. It is already bad enough that this is the case, and we should not make it even more true by adding to the nonsense. There can also be piece values that have a more solid basis, and I think readers should have the right to distinguish the one from the other. So as far as I am concerned people can publish anything, as long as they clearly state how they arrived at those values. Like "personal experience based on N games I played with these pieces" or "based on counting their average number of moves on an NxN board" or whatever. If there is a non-trivial calculation scheme involved, it is fine to publish that as a separate article, and then refer to that.
The problem with intuition is that it is notoriously unreliable. Humans suffer from an effect called 'observational bias', because one tends to remember the exceptional better than the common. This is probably the reason that GMs/world champions have grossly overestimated the tactical value of a King (as ~4 Pawns): in games where the King plays an important role it can indeed be very strong, but there are plenty of cases where a King is of no use at all (because it cannot catch up with a passed Pawn). These tend to be dismissed, as "the King played no role here, so we could not see how stong it really is". While in fact you could see how weak it was by its lack of ability to play a role. In practice two non-royal Kings are conclusively defeated by the Bishop pair, (in combination with balanced other material, and in particular sufficiently many Pawns). In games between computer programs that most humans could not beat at all. Of course none of these GMs ever played such a game even once.
Other forms of intuition often result from application of simplistic logic, rather than observation. It is 'intuitively obvious' that a BN is worth several Pawns less than RN, as B is worth several Pawns less than R, and it is their only difference. Alas, it is not true. They are almost equivalent. It ignores the effect that some moves can cooperate better than others, and in games BN + Pawn would score convincingly better than RN (and on average even beat Q).
We should also keep in mind that piece values are just an approximation. It is not a law of nature that the strength of an army can be obtained by adding a value of individual pieces, and that the win probability can be calculated from the difference between the thus obtained army strength. And indeed, closer study shows that it is not true at all. The win probability depends on how well pieces in the army cooperate, and complement each other, and how effective they are against what the opponent has.
For example, A=BN and C=RN are more effective against a Queen than against a combination of lighter material (say R+N+2P) that in itself would perfectly balance a Queen. Because all squares attacked by the latter, even though very similar to the number of squares attacked by a single Q, are no-go areas for a C or A, even when they are protected, while they would not have to shy away from a Q attack in similar situations. This causes Q+C+A < R+B+C+A, in Capablanca Chess, even though Q > R+B as usual. The extra C and A on the Q side are effectively weaker pieces than their counterparts on the R+B side, so much that it reverses the advantage. An extreme manifestation of this effect is that 7 Knights easily beat 3 Queens on an 8x8 board. Something that cannot be explained by any value for N/Q that would make sense in a context with more mixed FIDE material.
Your claim that B+2P ~ R and N+2P < R, which I don't doubt, cannot be used to conclude that B > N because of these subtleties. Piece values are not defined as how well the pieces do against a Rook, but by how well they do against a mix of opponent pieces such as these typically occur in end-games. And I have no doubt that the average performance of the Bishop suffers from the fact that there are many cases where B+2P ~ B+P, while N+2P would have done much better (namely when the Bishops are on unlike shades).
Note that the claim lone B ~ N was not based on what I would call a 'computer study'. I have no doubt a computer was used in the process, but just as an aid for quickly searching a huge database of human GM games. Not by playing computers against each other. The fact that a computer was used thus in no way had any effect on the conclusion. In the Kaufman study the claim was detailed further by stating that the B-N difference correlated with the number of Pawns, and exact equality only occurred when each player had about 5 Pawns; for fewer Pawns the Bishop performed better, for more Pawns the Knight. It is also common knowledge that Knights typically perform poorer in end-games where there are Pawns on different wings than when all Pawns are close together. This is of course also something that transcends piece values, which are defined as the best estimate for the chances without knowing the location of the pieces. Piece values are not the only terms that contribute to the heuristic evaluation of individual positions.
But to come back to the main topic: I don't think it would be a good idea to dismis any form of a quality standard on published piece values because "people should know that they should not believe what they read". That is an argument that could be used for publishing any form of fake news. It is already bad enough that this is the case, and we should not make it even more true by adding to the nonsense. There can also be piece values that have a more solid basis, and I think readers should have the right to distinguish the one from the other. So as far as I am concerned people can publish anything, as long as they clearly state how they arrived at those values. Like "personal experience based on N games I played with these pieces" or "based on counting their average number of moves on an NxN board" or whatever. If there is a non-trivial calculation scheme involved, it is fine to publish that as a separate article, and then refer to that.