GPS timing and computational complexity
Computational complexity lies in the correlator banks you need to despread the GPS signal, not in the position and time calculation (these are not separate things. You can't have one without the other). In fact, the TDOA system is very likely far more computationally intense, since it needs to do more cross-correlations.

You need to use the first method; 2. can't work, you'd need a clock that is about as stable and thus as large, power-hungry and expensive as that on the GPS satellites. Even with that clock, it wouldn't work well, because you still need to account for a lot of effects, including special relativity, that you can't solve via simple TDOA.

Computational complexity is still rather high, indeed, but really, not a problem since the late 1990s, where highly integrated chips for GPS became the norm. What it does do is drain the battery of your smartphone!