下に置いたカードが戻ってくるのはいつか

Because what goes underneath goes underneath in order. That makes the deck a queue rather than a shuffle, and a queue has positions you can count instead of odds you have to guess at.

A card you put under now sits below everything already there, so it comes up once those have been drawn: the deck size divided by what you draw each turn, rounded up.

Far less than it feels, once the deck is shrinking. The queue in front of the card gets shorter while the card waits, so a card put under later catches up with one put under earlier and they arrive together.

With forty cards, drawing four and putting one under each turn, everything put under on turns two, three, four and five comes back on turn twelve.

When the deck stops shrinking. If you put under exactly as many as you draw, the queue never gets shorter and each card waits the same number of turns as the one before it.

Then the arrivals spread out one per turn instead of bunching, and choosing the turn really does choose the arrival.

Anything that reorders the deck: a shuffle, a search, a card that moves others from the bottom, or drawing an extra card outside the normal turn.

It also treats every turn as identical, so a turn where you draw more or put more under has to be read as a different line rather than fitted into this one.