Something I see in the gym every day are private coaches (usually from the gym itself) being paid to walk around and give some basic exercises to the trainee. Most people walk into coaching with a simple assumption: I have a goal, and the coach is paid to get me there. But that is not true. The coach is also running a business, and businesses respond to incentives whether the owner is transparent about them or not.
If a coach is paid by the session, by the package, or by ongoing supervision, he is not automatically rewarded for making you get results quickly. Specially not to tech you properly how to train on your own. You wouldn't do that as a coach either right? He is rewarded for keeping the relationship entertaining, avoiding injuries and complaint, and getting along with as many clients as possible.
Coaching never solves the fitness problems you really want solved. The client gets endless corrective work, progress that is always deferred to the next block, and just enough added complexity to make independent judgment feel irresponsible. They get a lot of resting time, "we don't want to make the session be too unpleasant", a lot of exotic and interesting exercises "because we don't want the session to be boring" and mostly chatter so we build relationship. Look, that's fine. But if you want results, that getting you nowhere mate.
This is easiest to see in fitness because results, when you do things right, usually come fast. The client wants to run faster, get stronger, lose fat, or become more competent on his own. The coach is often rewarded for keeping the process supervised, carefully tiered, and permanently in need of another assessment, another correction, another package, another block. Always relatively interesting, comfortable workouts. Look, real, effective workouts are boring. Youtube any famous bodybuilder and, if you find their actual workout, you'll see that it's just loading a large variety of basic exercises with a lot. It's boring, it's simple. It's not easy! It hurts a LOT and takes a lot of willpower to get through a proper one.
Injury risk
Look, you're using your body. You're going to get injured at some point. You should strive to avoid it, but you will. It's ok. I've gotten stronger through injuries, forcing me to change exercises and learn something new while recovering. My shoulder injuries used to hurt like hell, but I got so much strength training done to recover and avoid them that my shoulders feel bulletproof now. Same with hips, legs... You injure, it hurts, you recover and you build up your body. You do need caution, but don't pay someone a shit ton of money to stand next to you and give you basic exercises that you can google yourself.
This is not a rant of private coaches. More on their usual incentives. The gym pays them to have trainees and keep them entertained. Not to get results. Hire a coach for a competition? That's a different thing, now they're incentivized to help you win that tennis championship, that Jiu Jitsu tourney... because it makes their business look good and others will come as well. Specially when they get their skin in the game, when they get a percentage of your winnings. Private coaching is a critical key for athletes, you NEED it so you can focus on your sport while trusting your coaches with your nutrition and training. But for normal Joes (and Janes) with office jobs? Do some google searches, read an ebook for a couple of hours and you're good to go. You'll get far better results.