When I was a kid, I thought this saying was just a complaint about the quality of cooks in general. It had an air of resignation to it: "You know, there's just too many cooks out there, who can't do broth worth a damn."
I never once got the mental image of a whole bunch of cooks, working simultaneously on that broth. The saying doesn't succeed, from an evocative, descriptive standpoint. If I were to say to you "too many quarterbacks throw interceptions," would you think, "ah yes! When a whole bunch of quarterbacks all get together to try to make one throw, it's bound to go awry!" No, probably you would not (albeit: true dat, one supposes).
It took forever for me to get that they meant a whole bunch of cooks, all working on the same broth, and spoiling it as a result. I have to ask: couldn't they have picked a slightly more complicated dish than broth for this saying? It doesn't seem remotely feasible for all these cooks to be in there on one broth!
And don't say that's the point because that isn't the point.