The issue would of course be that our trusty protagonist fighter or thief dual class character has to swap over at exactly the right level. Otherwise they're just wasting all that juicy experience. Now can you imagine the horrow of having a level 7 fighter that actually reaches level 8 before he can find a chance to become a bookworm? That would be a complete disaster. Crossing at level 7 works, crossing at level 8 is totally horrible!!
Joking aside, it wasn't entirely bad how it happened in Planescape Torment, but obviously it might be a bit hard to make this work with the entire class list of 5E, rather than just fighter, thief, and mage, which is what PST had. But it would be pretty cool if one could actually "talk shop" about class abilities with other party members of the same or a similar class. Halsin is a big time mega-druid, why can't he inspire our bear form? Laezel is bigly into swords, virtue of her being a gith, so why not reach a point where she gives a +1 to hit with githyanki swords buff to both herself and the protagonist? And so on, and so forth.