Obviously not the same thing as having one "On Stage".
and honestly, How often does that scenario happen?
For me, fairly frequently.
For example, a couple of years ago, I was playing a revival of a show called Closer Than Ever. On opening night, I was sitting in the green room before the show. The music director/pianist was sitting at the table, writing something. The stage manager called places, and we started up the stairs to the stage. As we were halfway there, the MD handed me the sheets he had been working on, and said, "here, we are going to play this tonight for the exit music." So we did. No problem.
Earlier this year, I got a call to do another theater gig. It was a workshop being conducted by some Broadway performers for high school kids. Band was again onstage: piano, drums, and electric bass. The gig consisted of a 2 hour rehearsal followed immediately be performance. (I think that the rehearsal was 5:30 to 7:30 and the performance was at 8.) No music was provided in advance. We read down the charts once or twice at the rehearsal, and then played the gig. Again, no problem.
I have a gig next Saturday night for the grand opening of a new theater space. The band will be on stage, and I know that the performance consists of a half hour concert. But beyond that, I have no idea what I'll be playing. I'm sure I will be handed charts when I get there, and I'm sure that the performance will go well.
The following Wednesday I have an orchestra rehearsal for a show I will be playing throughout November. Then on Friday, I have a rehearsal for a concert production of a different musical I will be playing on November 3 through 5. In fact, on November 4, I have a matinee performance for one show in the afternoon, and a dress rehearsal for the other in the evening. There is no time, or need, to memorize the music for all of these different shows.
These are all pro situations. I don't dispute that there are situations where a music stand is unnecessary or even undesirable. But the argument that a music stand is always "amateur" or "uprofessional" is absurd.