The problems you point out might be related. Pointless shuffling of pieces in the opening often occurs because within its look-ahead horizon the position that the AI's evaluation considers best is not at the horizon, but can already be reached in fewer moves. The AI then deosn't mind to waste tempos in the effort to reach that position. It sees that it can be reached anyway, and it saw nothing useful to do afterwards. If advancing Pawns is considered good, there will allways be some Pawns to advance in the opening, as you have so many of those. In that case there will be an incentive to reach positions with good piece development as early as possible, as this will then leave you more Pawn moves to improve the score further.

There is one problem, though: mindlessly storming ahead with all Pawns will expose your King, and often other valuable pieces as well, as it would allow your opponent to creep behind your Pawns. Pawns really make an essential contribution to controlling the squares near your camp, preventing invasion by light pieces. So it is important to keep some back, both as obstacles for blocking enemy sliders as for the squares they attack. In Fairy-Max I managed to get reasonable Pawn play by awarding Pawn pushing in general, but then give a penalty if 2 files left or right from the square of origin of that Pawn there wasn't another (friendly) Pawn. Because that means no Pawn is left to control the square that the pushed Pawn used to control. So after 1.e2e4 the move c2c4 would be discouraged for creating a weakness at d3, while d2d4 would still be OK, because b2 and f2 are still in place.

The problem is that this is very much based on the Pawn move being like in FIDE/Shatranj. It makes no sense for Berolina Pawns, Shogi Pawns, Janggi Pawns etc. And there is no guarantee that the Interactive Diagram will always have Pawns that move as in FIDE.

All Pawn types are useful for shielding the King against slider attack, and this can be expressed by giving a penalty for pushing Pawns next to the King to compensate a general Pawn-advance bonus. The problem there is that the King often doesn't start at the location where you want it to be. So applying a penalty for pushing Pawns in the King shield in FIDE would just prevent opening with your center Pawns, and would make the engine push the b- or g-Pawns instead, spoiling its future King shield in the castled positions before the King could ever get there. Again, this is much dependent on the initial position and the type of castling, which can differ from case to case. In general castling can be assumed to be good (or there would have been no need to make a special rule for allowing it), so the AI gives a lot of bonus for castling. But (unlike Fairy-Max) it doesn't search deep enough to have castling within its horizon when all pieces are still on the back rank, and a bonus that it cannot see how to get will not affect its play no matter how large you make it. If it prefers piece moves over Pawn moves there is a better chance that it clears the path for castling, (and subsequently castles) before it destroys the Pawn shield at the castled location.

But again, this is dependent on a lot of factors, so it is not that simple to produce the right behavior. Perhaps something like giving extra bonus for moving pieces between King and corner away in variants with castling. But in FIDE-like setups those are basically all non-Pawns. And it would be pointless to do this on both wings. So perhaps there should be a penalty proportional to the number of pieces between a (virgin) King and corner on the side where this number is lowest, to not needlessly encourage quick development of pieces (i.e. before pushing many Pawns) on the wing where you are not going to castle to. Problem with many variants is that the pieces can jump over the Pawn wall. For FIDE only Knights can do that, and you have to push Pawns to develop the Bishops, even when you would prefer developing pieces over pushing Pawns. It is not easy to get a good balance between Pawn moves and piece moves in all cases.