Can a defective starter stop my car?

Just before christmas my car died while my girlfriend was on her way to work.
She told me that when the car stopped it went completely silent (no engine/lights/music). Since she drove for about 15km (mostly highway) before that happened I suspected a problem in the realm of generating electricity. Bad smell made me fear for a more expensive defect of the clutch.

Fast forward and the nearest mechanic (to whom the car was towed) called and said the car is running again (with no work done), but the starter is in bad shape and he asks my permission to replace it since it might be related to the problem.

If my usual mechanic asked me that I would say "if you think the car is still worth it, go ahead…".
But in this case I don’t 100% trust the guy since the whole thing looks a bit shady to me (for example: he said his workshop could be found on the internet, which I couldn’t(even looked up a company index of the town) And that’s not the only thing)

So my question is:

Is it pausible that a defective starter stopped my car?

My googlefu brought less than satisfying results. Most sources I searched simply pointed to a faulty battery if the starter won’t work. Only one site ambiguously suggested that such a defect could cause other issues, but nothing so far detailed what that could really be.

My main fear is that

1. the mechanic simply didn’t find anything and wants to do something so the invested time isn’t wasted and
2. the car simply works again because he charged the battery and I will have a stopped car when I arrive back home (altough a mechanic should notice the battery light if that is the case i guess…)

The car is a Toyota Corolla with 20 years and around 150.000km

I would prefer answers that go beyond yes/no or trust/don’t trust that guy. I would say I only have an average understanding of cars, but when an issue arises I love to know as much about it as possible (which occasionally annoys my mechanic…)

Thanks in advance