Theoretically, yes...
However, moving the start switches to FLT is easier to use (two switches being used instead of three), easier to remember as well as you have to consider the fact that if you start the memory items quite soon and obviously depending on the type of engine failure, you will probably still have both generators supplying the electrical power to all the busses, so relight with 2 ignitors is much more likely than with a single one.
Additionally, 737 can depart with R igniter U/S as per the MEL, whereas then the L igniter has to be wired to the AC standby bus.
I think moving the start switches to FLT is just bulletproof solution to cover many possible scenarios and to ensure the highest probability of a successful engine relight.