Mechanic is saying rear differential is shot and needs replacing. Car hasn't been making any noises, actually only took it in to look at a leak in coolant reservoir. Wouldn't there be lots of grinding noise if my differential was bad?
TL DR: Yes, noise would most likely be present if the differential is going/gone bad.
There are two main factors (I can think of) which would indicate a differential issue. First, if there is noise (as you stated). You'd hear a whine, usually on acceleration or deceleration. Mind you, some differentials are noisy in the deceleration portion and will live that way happily for many years. When they are going bad, you'll hear a change in noise level, where they will start getting louder over time (in either or both directions). The second way is if there is a leak. However, a leak in and of itself is not indicative of the need for complete replacement, but could be an indicator of bad things coming ... or could just mean it needs a new seal or gasket. The last thing you want to do is run the differential dry, as that will lead to imminent destruction.
In some shops, if a mechanic spots something which needs to be fixed and the customer buys off on it, the mechanic will get a higher percentage of pay on the repair. So, if the mechanic normally gets 25% of the per hour rate to fix an issue which comes to them, they might get 40% if they find the issue. It's lucrative for them to do so, so they are on the lookout for other issues. The problem is, I don't see how the mechanic could tell there's a problem with the differential either. In your case, it is buried within the transmission (as a transaxle). While a mechanic will look for other things while doing their assigned task(s), they aren't going to spend all day looking for stuff which doesn't exist, either. If they aren't working, they aren't making money ... looking for non-existent problems won't get them anywhere.
All-in-all, if it were me who were in your situation, I'd either not worry about it (due to the lack of noise) or if it's bothering you, take it to a different shop. To me, your description makes this sounds like a "Pink Tax" situation. I hope that isn't the case, but it sure sounds like it.