If you have never seen a trap gate/medina hinged gate, just think of the bulls at a rodeo. The riders still put on the riggings and do everything the old fashion way. The gate swings out to release the bulls. This method has been used for many years and goes back before we had all the modern equipment.
There are many things that I still do with a medina gate. It is just simpler and faster when you are working alone for some minor things. Older cows are not as nervous. If I am just spraying cows for flies, I'll do it all with the medina gate.
If I am going to inject 2 cc's of penicilin into the eyelid of a calf/cow along with a separate injection of LA200 to cure pink eye, and it is the only animal I am working that day, they are still going to go into the squeeze chute just because I have it. When I was a child I remember dad and grandad running animals into wooden chutes or trap gates and then snubbing them with a rope. I also remember broken boards, rope burns, skin losses and almost killing a few.
I've also seen some cows drop in a squeeze chute and people not being able to get them out of the head catch. Dead animal and it is just a freak thing that happens.