I am not sure in 5e, but they certainly did in 3e. Mirror image was a great way to buy yourself a few rounds of safety for example.
Mirror Image isn't a particularly good spell because in general, if you can spend a round casting a spell to protect just yourself, either Glitterdust(offensive) or Invisibility(defensive/utility) are likely fantastically better choices. Particularly in the context that Inspire Courage goes on for 5 rounds after you do it, so it is no big deal if you want to be invisible, yet have your friends benefit.
One of the big things here to emphasize:
If you are going to spend a round making it hard to hit you, there needs to be one heck of a reason why you just wasted a round convincing your opponents to leave you alone to focus fire your friends.
In general, the big issues here sound as if your table didn't have a rule 0 discussion. Namely, what's everyone's expectations. Is there any kind of thing that's going to be a big problem for anyone else. There's nothing wrong with a PC who has low defenses provided it is that kind of campaign. You're strongly suggesting, in practice, it didn't play out that way. i.e. if you think, as DM, that a PC isn't going to make things easy either for the player or you, you talk to that player about what your expectations are and ask the player how that PC helps you match those expectations.
A lot of people think 4e was about downgrading the DM from being the ultimate authority figure, but the DMG is full of advice about how to get everyone onboard with your campaign. So you don't actually need to put your foot down as much, because your players understand what your goals are.