Not anywhere near as much a problem as you think. I have done this in my emergency services dispatch job for the past 20+ years, and it's as I was trained by people who had 20 years in when I was the greenest rookie in the building. I'm also a firefighter, and if I hear a dispatcher say "01 Engine, go ahead" when 01 Engine never called them, and then "01 Engine, roger," a few seconds later, it tells me they're busy. I don't get on channel and say "hey hyuk dispatch yer talkin on the wrong channel", I know that they're talking on *many* channels at once to save time. Not only do they not have to pick the channel to respond on, but users of the channels know that the dispatcher is busy on another channel.
I multi-select all the channels I'm responsible for and then I can use my foot pedal to transmit, freeing up my "mousing hand" normally used to scroll and pick radio channels and xmit on them, for typing properly on the CAD, making things that much faster. I don't have to move the cursor on the radio and switch channels with every transmission, and can type a full remark with touch-typing quickly while the person is talking to me, rather than waiting, keying up with my one hand to "roger" them, and then waiting until I'm done that to type the reply.
ATCs and other air controllers may not be doing exactly the same thing. Like others have said, sometimes one controller is handling multiple zones or areas of responsibility, which is similar to my situation mentioned above, but other times, there may be multiple frequencies used for the same area. I don't know if America does it the same way my country does, but for example here civilian air is on one band and military is on another, but one controller will be handling both. So the controller will be talking on both VHF and UHF simultaneously, but the replies will come from only one band depending on who's answering him/her.
Lastly, on one of the things you mentioned in the post immediately above this one - nothing typically gets missed, or if it does, it's recognized immediately and the dispatcher/controller asks for the person to say their message again. Radio consoles mix all the voice traffic into a single stream of audio that is played into your headset - it's not like a scanner where only one signal can be heard at one time. Sometimes the console audio is advanced enough that you can route one signal to your left headphone and one to your right, or similar. In my center, and my experience, dispatchers are skilled enough to pick out 3 or 4 conversations going on at once. Not constantly, but if three people managed to key up at the same moment and say something to me, there's a good chance I would be able to recognize each transmission and understand what was being said. I wouldn't let them continue that for very long - lots of "stand by" would be applied while I untangled that bowl of spaghetti. But what I'm saying is, for one or two transmissions, if somebody blurted out something, it wouldn't be impossible to decipher what was said, and it'd be rare that something was missed. Even if it was, the person who didn't get a reply to whatever they said would 9 times out of 10 point that out and call again.