I encountered this same issue so I did some testing and here's what I observed:
When connecting to HOME, all Pokemon regardless of whether or not they are actually moved will have all PP set to 0 and their Height and Weight adjusted to be equal to the Scale value (e.g. a Pokemon with Scale 137 will have both Height and Weight set to 137). This happens even if you immediately save and quit without moving any Pokemon. I don't know if there are any other changes, but these are the two issues that violate PKHeX's current legality check. The HOME tracker is still all 0s for Pokemon that were never put in HOME.
After a save file is accessed in HOME and you start up the game, PP can be corrected under certain conditions. I haven't tested this extensively, but it seems to be automatically corrected for Pokemon in your party and for Pokemon in boxes if you alter their moves such as swapping move positions. Even if you swap just two of a Pokemon's moves, all their moves will have their PP fixed. But Pokemon that don't meet these conditions will still have 0s for their PP.
In PKHeX, if you change a Pokemon's PP back to their normal values and set their Height and Weight back to 0, it will pass the legality check. If you then save and restore that save file, access it in HOME, and then open that save in PKHeX, that edited Pokemon will again have their Height and Weight set equal to the Scale value and PPs all set to 0. I don't know if editing a Pokemon this way will have any repercussions though (I used a separate basic account of HOME in just in case).
After a save has been accessed by HOME, newly-caught Pokemon in Z-A will still have Height and Weight at 0 and non-zero PP values as long as HOME is not accessed again.
The takeaway is that HOME seems to be altering certain values for all Pokemon in the save regardless of their origin and whether they were moved in HOME or not. The Height/Weight change is probably for future-proofing those values. iirc Legends Arceus had the opposite problem since Scale didn't exist back then, Pokemon transferred from that game simply had their scale set to 0 until a later HOME update. The PP issue might just be HOME ignoring PP values for Z-A (since the game doesn't utilize it) and just sets it all to 0 when it alters the save, especially since the game corrects it anyways under certain conditions. I haven't gone as far as comparing pa9 files for other possible alterations though.
