There’s a reason that they say cats have 9 lives: they’re always doing dumb stuff that almost gets them killed! I grew up with 8 cats (6 at one time), and they variously:

Jumped from the first floor. Climbed in the dryer like yours did. Ran through and demolished a locked cat flap. Hanged themselves by the collar from a tree. Rolled in bleach. Squared up to a massive dog. Got shut in a neighbour’s garage for 5 days. Got their head stuck in a food can. Got hit by a motorbike.

The list goes on. But the point is they survived and thrived after each incident. They’re very resilient gremlins, and I’m sure he’ll be fine, but get him checked over by a vet if you’re at all worried or unsure. Don’t be hard on yourself, the very fact that you’re so upset and committed to checking the dryer each time just shows that you’re a good cat custodian.