O(1) doesn't mean constant, it means bounded by a constant. An algorithm can be faster with small n and converge to a horizontal asymptote as n goes to infinity, and it would still be O(1).

In a real machine there is no infinity, but hundreds of GBs of memory are "infinity enough" compared to the cache size [1]. So asymptotic analysis is still a decent model.

I'm surprised that even a CS professor confuses this.

[1] Ok if we want to be pedantic memory access is logarithmic due to the traversal of page tables, but you can use huge pages.