If you like riding "robots" request "robots". You are the customer you are likely to be paying a fairly substantial sum, they need your business, if you like riding Fred or Sunny when you book your lesson tell them.
Where I ride it's a joke between me and the staff who work in the office that 95% of the time I ride the same horse, sometimes they momentarily pretend they've put me on someone else just to wind me up! My favourite horse and I have over the months developed a great relationship, even though she's also ridden by loads of others she clearly remembers me and we have an understanding. We practice various movement and every week we both improve. This week she went so well one of the other instructors didn't recognise her asking my instructor who she was? I was chuffed to bits I understand her and she understands me, I know how to get the best out of her. I'd come from a very stressful day at work and thought I was probably too tired and wound up to enjoy the lesson, I finished feeling exhilarated. This is what it should all be about, you should be enjoying it. Having said this riding will never be easy simply because it's not easy, I've spent 40+ years on the back of horses I used to ride 3-4 hours a day, I've had 1000's of lessons and sometimes I'm left stunned by my own incompetence! We do learn from horses that don't suit us but do we need to learn what they are teaching us if your just doing g it for fun. I recently rode very badly an advanced dressage horse, popular with any clients at the stables I go too, ok I improved my flying changes but I wasnt really enjoying it. I didn't like the way it moved, I couldn't steer the thing, or even get it into canter at times, I finished the lesson demoralised at how badly I rode and wondering if I'd made the right decision coming back to riding after have had a break.
As I said above if your current stables won't play ball find another one, or how about an intensive day(s) of riding at either your stables or somewhere different, I can highly recommend where I go. Where I ride lots of people come for an intensive day(s), they've obviously spoken to the office and have set objectives whether it be to competently walk trot and canter, or learn to do piaffe, the staff work hard to help customers achieve their objectives. Perhaps this would boost your confidence and move your riding on a bit.