Do you think a big part of the problem is coaches can't communicate well how they could fix the FH, BH, etc for a player?
I think a much bigger problem is students not properly practicing what they have learned. Especially players like the OP who have "wrong" muscle memory to unlearn which they have been building up for years and years.
I have been "unofficially" coaching a few friends over the winter, mostly with some general tips etc.
One of them has 2 main problems: bad grip on his backhand and a "waiter tray" serve.
Neither are easy fixes. But I'm certain the backhand is easier to fix then his serve.
To "unlearn" the bad grip and "relearn" the good grip, I bet he could do it in 10 to 20 hours of practice. His footwork is ok, his forehand is ok and his body movement is ok. It's really just the grip.
His serve however... that thing needs to be deconstructed completely and rebuild from scratch. 10 hours will not suffice. We're looking more at 50 to 100.
The reason I mention this, is because it doesn't require 10 to 20 hours to explain him proper backhand grip. Nor does it take 50 to 100 hours to explain proper serve technique.
Showing / explaining him proper backhand grip takes just 5 minutes. It takes 10 to 20 hours of
focused practice to implement it. Same for the serve.
I think that in general, the problem is not so much the coach who isn't communicating well but rather the students not putting in the required practice outside of the lesson court to put into practice what they've learned.
If my friend gets a private coaching lesson to fix his backhand grip and goes to court with friends the next day and just returns to his usual bad grip... that's not the coach's fault.
Whenever I see people take lessons and not progress, that's usually my observation... the students not putting into practice what the coaches have taught them... And just returning to their bad habits.
Say, I don't mind paying hundreds of dollars to a coach if by 6 months I can get a fundamentally decent, low-power serve.
Sure. But as said, the coach doesn't have a magic wand. It's still you that's going to have to put it into practice. And 90% of that practice, will not be during lessons.
Back in the day I took private lessons to play drums. It was rather expensive at 75 bucks per lesson. That drummer had a busy life also and was not at all interested in people who took lessons "as a hobby". He wanted to see that his lessons were put into practice. So his attitude was to bombard me with exercises (sticking patterns) and techniques during 30 minutes and then told me to not come back until I could comfortably do everything he just taught me.
His idea was that the lesson was for theory, tips and technique. Not for practice. Practice was for on my own time.
Tennis coaching at the club player level shouldn't be much different imo. Not necessarily with the "don't come back until...", but the overall idea I think is very healthy. Why pay a coach to just do drills to implement what you have learned? You can - and SHOULD - do those drills on your own time.