egg wash vs milk wash vs nothing - does it actually matter?

There's a system, and "nothing" is a legitimate member of it - plenty of course breads bake without any wash at all.

Egg wash = the shiniest, deepest golden-brown. The protein browns and glosses; it's the patisserie finish for enriched breads and anything that should look lacquered.

Milk or cream = a gentler, softer gold with no shine. Homely rather than glossy.

Butter, brushed AFTER baking = doesn't colour anything, but softens and perfumes the crust. The pav move.

Nothing = the crust the dough earns on its own - right for rustic and crusty styles where you want texture, not gloss.

So it's a finish decision, not a requirement. Decide if you want shine, soft gold, soft crust, or honest rustic - and wash (or don't) accordingly.