Well, the problem with the "gotta play without paper" idea is that it comes from a time when guys in their 20s lived in NYC, four to an apartment, lived on peanut butter sandwiches, and did NOTHING but play music all day long; AND they had a restricted repertoire. In the golden days of jazz, say 1928-1955, you played all the tunes your set of musicians played. So in '55, it would have been all the bebop favorites - probably a hundred tunes.
Now, most of us have to work for a living. Even if it's in music, it's teaching lessons, or band directoring, or writing arrangements, or sound engineering.
And most of us don't really want to do the four guys in an apartment, nothing in the fridge but a couple beers and some ketchup, living off the kitty.
And then, there's just SO DAMN MUCH music.
For me, for example, as a saxophonist and bassist, I'm expected to be ready to go on jazz from 1917 to last week, rock and roll from Louis Jordan on, country and western from the Carter Family to Blake Shelton, bluegrass, folk, etc., etc., etc., - it's thousands and thousands of tunes.
What makes actual sense is to have a quick reference, so I can look at a lead sheet, or chord chart, or hell even just some handwritten notes like "In Eb, first chord is the 4, first note is G, rhythm changes on the bridge". The much-derided supposed phenomenon of "head in stand, not aware of anything going on around him" doesn't apply to me, or to any of the other experienced musicians I play with who look at a chart. We see how it's laid out, and once the head's played, look away and play the music; but it prevents train wrecks.
Sure, if you are going to a jam session where the model is that you get up, call two tunes, play on them, then sit down, yeah, it's not unreasonable to have them by memory. If the model is different, like almost always with me, where you don't know what the heck is going to get called next, then a reference is handy. Even then, of course, there's going to be tunes that aren't in the book - but at least you reduce the number of trainwrecks.