I wasn't comparing scientists running a simulation with mathematicians running a simulation. I was comparing scientists collecting evidence that might disprove their theories with mathematicians running a simulation - because such a simulation collects data that might disprove their conjectures.

What is an example of something a scientist can prove with 'just logic'?

We'll need to agree on a subject who is a scientist and not a mathematician. The easiest example for me would be to use a computer scientist, but you may argue that whenever a computer scientist uses logic they're actually functioning as a mathematician, in which case the dispute comes down to 'what's a mathematician'.

In the event you don't dispute, I'd note that a lot of computer science has involved logic regarding, for instance, the nature of computation.

In the event you do dispute the status of a computer science as science, then we still have an example of scientists performing mathematics when possible, and really physicists do that too (the quantum formulas that don't mean anything are a fine example, I think). So, to go back to my original point, it's not like an accusation of non-elegance has to come from nowhere; those physicists are undeniably practicing math, and elegance is important there.