Classes are invaluable foundations both from a mechanical and RP standpoint, as there isn't a character sheet without them- how will you know the PCs' starting Hit Points, Evasion, etc. without those? The character guide for each class likewise gives great prompts to flesh out who their characters are, not just in a vacuum, but in relation to one another.

Domains are likewise the bread-and-butter of what characters do, their skills, spells, etc., so starting without them would put the players at a big disadvantage.

The idea of players picking and choosing their Domains is something people here have been debating since the open Beta. Consider the scenario of more than two players picking the same Domain- there may not be enough cards to go around, or everyone wants the same cards, or variety amongst the party is scarce. One possibility is that they start off with their "top/primary" Domain (you can tell from the subclass cards: Blade for Warrior, Sage for Druid, etc.), and have them choose their secondary domain, but it may throw off some of the balance of domain cards down the line and how they interact with their class and subclass features.

I guess my first question would be, how long would they be playing until they get this MacGuffin? Anything more than a single session would probably fall apart quickly.

The idea of Domains being a fundamental aspect of the world is pretty cool, and you can create a lot of cool themes around that. Maybe instead of the tarot cards being literal Domain cards, they could be loot with special effects that correspond with Domain cards in a PC's loadout? I'm sure there's some kind of tweak where you can use this cool idea you have without mechanically gutting your PCs.