In reply to Kirill:
> I get your point about athletes benefiting from coaching. But in my case I am afraid that the coach will just take one look at me and will tell me right there and then that I am hopeless. And that would be very difficult for me to get over.
No good coach would do that. Apart from anything else, they won't get any business if they do!
They would be saying try doing this instead of that.
A suggestion for you is to work on your footwork and make sure you are using your legs to push up, not your arms to pull up. Find an easy slab, with some 3s and 4s and ignore the routes. Using any holds of any colour to start with, climb to the top without using one of your hands at all. You can use that arm for balance, but that hand does not touch the all. Progress this by doing the same with the other arm, then by doing a very easy route this way.
Another thing to try is climbing with your eyes closed. It helps you feel the movements. Again, try this on an easy wall.
Still on an easy wall, big holds, climb up a little way, find a comfortable place to stand without holding on. Without putting your feet on any other holds than the 2 you are on, move about, see how far you can reach up, sideways, down, see how many ways you can move your legs, find some hand holds and stand on one leg, swap your feet and generally see just how big your range of movement is.
When you have done some of these, find a route that is maybe an easy 5. Something easy for you. Climb it as beautifully, as smoothly as you can. If the wall isn't busy, climb it again, but this time, pretend you are a ballerina! Or climb it again pretending to be a small excited child, as fast as you can.
Going back to something easier might seem odd, but by making it easier you can refine your skills and then take them back to push on through something harder.