A foreign object probably got flushed and is stuck in the base of the toilet. You need to remove the toilet and check from underneath.

Wouldn't the snake have hit it?

Sure, but I suspect you're not dealing with a blob that needs to be broken up, but rather a piece of plastic that's just getting move around, then resettling near the hole. You could try a few more times and maybe break the thing and get it going. BTW, I once had the problem you describe and it was a shampoo bottle cap that was stuck in there.

If you have a wetvac, this is a long shot but you could try it: pour a bucket of water down the toilet (to get water beyond the trap down to the object), then immediately suck all the water out. You might get lucky and suck the thing back out out with the water. Again, it's a long shot, but little to lose if you happen to have a wetvac.

I got 3 suggestions for a very slow flushing toilet, so I combined them and it worked. Then a couple years later it worked again. This was when plungering with a large plunger did not work.

1)hot water
2)soapy water (dish soap like Joy).
3)presure from the sudden dropped water

So I Stood up high with about 3 gallons of hoat soapy water in a pail and dumped it suddenly in the bowl. Nothing happened, so I bailed out and tried again. Once more and the 3rd time it worked!

Glad you mentioned the plunger. That's the first thing to try.