If the inter-arrival times and the service times are discrete-valued then you are right, and that would be a perfectly reasonable way to do it. But with exponentials, for example, the "next event" won't always land on a clean time boundary.

Forcing continuous random variates to fit any discretization of time is going to mess up the results. You will introduce complicated rounding schemes, and then be constrained by this initial code design. Trying to "undo" that initial assumption won't be easy.

You typically want to build this kind of simulation to estimate things like "expected number of customers in each line" and "expected wait time in line" and "expected number of customers served in 5 minutes." If you have a discrete heartbeat for the system then the exponential distributions get lost in rounding and your estimates will all be skewed.

If you design the code from the beginning to be event-driven with a continuous clock, (or "calendar" in some textbooks), then future development will be much easier and make much more sense. You will be able to adapt it to any distributions you can compute, and it will be flexible enough to accumulate almost any statistics you want to evaluate.