I don't play in an orchestra, but I'd guess that classical musicians are quite capable of reading their sheet music at the same time as looking at the conductor and listening to what's happening around them...
As for jazz jams, I'd rather play from memory if I can, but tunes are bound to be called which I don't know at all or which I've heard but don't know the changes. If I glance at iReal Pro then I can get a head start on following those changes whilst still paying attention to the rest of the band, improvising etc. to an acceptable standard for the context. The rhythm section not knowing a piece is not usually considered grounds for skipping it and we'll move on to another call only if the horn player(s) don't know the head, so I will have to play tunes I don't know. Being able to quickly transpose for a singer is invaluable (but see below).
The OP mentioned "Before each song, there is a clustef__ of people asking to be sent the chords," which is a fair point, I think. In my experience the delays and faff seem to be caused mainly by the following:
1. Difficulty operating the tablets, e.g. "what's 'AirDrop'? Do you mean 'AirPlay'? How should I turn that on?", problems connecting to the network, or emailing charts around, or someone using an Android device when others have iPads (a problem Google has been working on).
2. Singers not knowing what key they prefer to sing in, resulting in much fiddling with transposition to see which key might suit.
3. An obscure call which is not in iReal Pro or whatever real/fake book is present, resulting in hunting for charts on the internet or trying to get the caller to explain.
All three could be ameliorated by preparation before attending the jam.
Delay (3), in my experience, tends to be people attending a jazz jam whose interest is more blues, soul, pop &c. and whose call is not a jazz standard. Unfortunately, they often seem not to know what the form is and struggle to explain it - I have on occasion asked if the blues we're about to play is 8, 12 or 16 bars, for example, and they can't tell me. "You'll work it out, just look at my left hand" is all very well, but it takes a while and meanwhile things won't be sounding great.