That’s an interesting question. I appreciate questions like this because I also like to pick apart linguistic things like this. I had to think about this for several minutes, but I think I figured out a possible explanation:
First, like most colloquialisms, I think it's largely unconscious, but I think one explanation for why that subconscious language might have come about is that in this context, "got" is a more passive verb than "see."
In some contexts, the verb "to get" could be used to mean "going out and getting something," to seek out and acquire something, which is active. But in this context it's more like passively "receiving" - we didn't seek out the understudy, but it was given to us. We "got" one. Similarly, on the flip side, the verb "to see" could also be passive in some contexts, but in the context of theatre, it feels more active. Because you're putting in the active effort to buy an expensive ticket, going to theatre, and making an evening of it (same reason you "see" a movie in theatres, but "watch" a movie at home).
So basically, going to "see" theatre, or even "seeing" a regularly-scheduled actor, is all part of the same active choice, over which the person has control** But "getting" an understudy is something that happens to you, and you don't control it, so it's more like something you passively "receive" (aka "get" or "have"
**(obviously you don't ACTUALLY control if you see the regularly-scheduled principals, but because they do most performances, and because people often go to shows specifically to see certain stars, it feels to the audience member as if it's a decision they are in control of)
Updated On: 1/6/20 at 11:40 AM