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PKHeX is a save file editor; HOME stores its data server side. I'm not going to implement customizable save data formats to fake an experience, as everyone would have preferences (longer box names, more slots per box, moving groups of boxes, viewing in original format, storing as multiple formats). Sure it may seem nice for collections that are not small, but it's not really useful when your collection is large and spanning all games. When you need to grab something, opening up the pkmdb and searching (including the advanced filters) will quickly give you what you need. PKHeX even reads the backup save file data into the pkmdb, no need to dump files.

PKHaX just hides the legality check indication, shows all abilities (instead of filtering and updating the bits), and allows manual stat entry instead of automatic stat calculation. It's only really useful when trying to edit ROM hack save files (unsupported) or making dumb hacks like Wonder Guard stuff.