The underlying technology behind (almost?) all of these is TOTP (https://en.wikipedia.org/wiki/Time-based_One-time_Password_a...) which pretty much just depends on both systems having clocks that are reasonably close to synced. The initial value is basically a random number generator seed, and given the seed and a number of iterations (based on the time differential from a set starting point) calculating a code is fairly simple.

There would be problems on fully-isolated systems experiencing clock drift, but on any modern Internet-connected system using NTP or on any cell phone with time synced to the network it shouldn't be a factor. The most likely problem scenario is probably a corporate network using only an internal time source that drifts.

Doing a single code as validation only makes sense to catch transcription errors since in case of problems someone could end up locked out of an account.