I have a WW squeeze that was given to me. It is the least used piece of equipment that I own. The cows hate the thing. I can have all the work done I normally need to do on them in the chute in way less time than it takes me to get them into the squeeze. The biggest benefit is that the cows stay calm. AI is even better in the regular pipe chute. Just close the gate in front of them and put a pipe behind them. It's wam bam thank you mam. In a squeeze they get upset, some go to their knees, others just fight. Conception rates are far greater not using the squeeze. About the only thing I found a head gate good for is dehorning, or getting the halter on my spotter. The squeeze is used for branding, udder work, nut removal, and the like.
With that being said I realize there are cattle that need to be worked in a squeeze. F1 Brafored calves that were just weaned for example. I have worked cattle for the better part of my life with out the aid of head gate or squeeze. They are not mandatory, but in many cases they sure make life a lot easier.