In the real world, learning the profession of being a Soldier takes a few months, same for being a police officer. You can learn a lot in a few months of downtime.
D&D games I've played in have often had weeks or months of downtimes between adventures, especially between plot arcs.
Maybe they decide to take some downtime while their Cleric is working on crafting some big important magic item they need and spent several quests getting all the components for, and right as he was finishing up with that, his Church decides to send him on a retreat to a monastery that will also take months and in that time he lives cloistered off in the mountains and essentially gains the hermit profession. . .
The fighter the party got conscripted into the Royal Army, and there's a year long gap of his conscription before he's available for adventuring again, and in that time he learns soldiering.
The party wizard from a hermit background decides to join mainstream wizard society and joins a college of wizardry and enters academia. . .and he's essentially learning the Sage profession.
The party thief from a charlatan's background decides he's actually pretty dang good with crafting and figures he'll try his hand at honest labor with a crafting guild. . .and picks up the Guild Artisan profession.
Then, a year later when the Cleric has finished his crafting and pilgrimage, the fighter's conscription in the army is ending, the wizard now has a minor faculty position at the college and is now eligible to take a sabbatical to go on some more adventures, the thief is now a recognized journeyman and is free to travel to other masters to learn from (which means he can also meet up with his old buddies). . .and after a year apart and learning and having their own adventures they have each learned a new profession, but are now back together and able to adventure again.
Yeah. When I said I could see it being reasonable, I had in mind a game with adequate downtime for it to work. My games haven't worked out that way, but others' games can differ.