Why do so many gyms focus on functional training during Personal Training sessions?

I was around when the functional movement began more than a dozen years ago. The basic idea is to perform repetitions of exercises that mimic movements the human body performs either day to day as in “activities of daily living” (aka “ADL”) such as squats which we perform every time we get in or out of a chair, up to more complex movements such as doing a one-arm balance row on one leg which mimics actions like reaching out to pick something up off the floor that may have rolled almost out of reach, or someone who has a big dog that often pulls their owner, throwing them off balance.
Functional exercises also help improve balance and agility. If you have a client that is a recreational athlete, it will greatly improve their performance to mimic the sports motions from various angles with resistance. Using the ViPR for core chops has significantly improved one of my student’s tennis game. Jump squats help improve the performance of beach volleyball and basketball players. If it improves the functionality (or performance) of your clients’ or students’ movements, then it is functional training. And as mentioned above, helps prevent injury as the body is more prepared for the unexpected (that’s why balance exercises are so popular).
In basic terms, as a physical therapist once told me a long time ago (phys. therapy is where this whole movement began, incidentally), while I was rehabbing from knee tendonitis, “don’t waste your time on seated leg extensions, that’s not functional, your body never does that movement in real life, except in a gym on a machine…” In other words, there are exercises that just work the muscles from an aesthetic perspective, like dumbbell presses on a bench, then there are exercises that serve a purpose and may come in handy someday, like push ups. Both work your chest, but one is functional and the other is not.