Steep endurance climbing is more about the metabolic pathways you've developed and trained rather than strength, but getting stronger helps. As someone else said, if 5.10 moves no longer get you pumped then you have nothing to worry about at that grade. This is the ideal way to go if you want to improve past 5.11 and get into 5.12, 5.13, possibly even 5.14. The thing is each grade requires a little bit more from you. Any slob can climb 5.7. To climb 5.8 you need to have a little mental strength but generally fat people can still climb 5.8. To climb 5.9 you need to start getting some technical skills, most people can't climb 5.9 on their first day.
Climbing 5.10 requires some specific physical strength in the forearms, but almost nowhere else. The average person has a strong enough back, shoulders, and core to climb 5.10 provided they are at a healthy weight. Mental and technical skills also need to be improved to climb 5.10.
Climbing 5.11 is mostly about getting mileage as a climber, and being "reasonably fit". At this stage you should be at a healthy bodyweight, and have a decent amount of strength. You need mileage, and lots of it, to solve the new climbing puzzles you're going to encounter, and finger strength is going to take on a whole new meaning, make sure you're bouldering a lot to build that finger strength.
To climb 5.12 you need to build on everything you had for climbing 5.11, but at this point your mental game should be pretty flawless. If your body is capable of doing the route you should be doing it, your mind should not be holding you back anymore. Your technical skills as well should be pretty well filled in, there are always new things to learn, but at least from a sport climbing perspective you should be able to do 99% of the moves you encounter with technical proficiency at this point. Finger strength again, is key, and you should be bouldering often to maintain and gain finger strength. Body composition matters a lot more at this point. At 5.12 routes start to get very steep and your fingers can only haul so much weight up the rock. Get rid of the beer gut.
To climb 5.13 you need to start thinking of yourself as an athlete. It's no longer good enough to be "in shape" or at a "healthy weight" you need to be at an athletic weight now. You need your body fat to be low, sub 15% but sub 10% would be better. You need to have stronger upper body and core muscles now than the average climber. Not just the average person. You need to start specifically training those muscles if you haven't been and you didn't start out ridiculously strong. You need to be fingerboarding now to force finger strength to improve, you need to be campusing to work on power. Climbing improvement at this point comes down to training.
To climb 5.14 you need to forsake any thoughts of climbing being "just a hobby". Climbing is more than just a hobby if you climb this hard. It's your sport, and probably one of the most important things in your life. Before you eat anything you think of how it will affect your climbing, the amount of time you sleep, how you train, when you train, all of it is calculated. Progress is hard to come by and standing still is no longer an option. You're always moving at this point, and if you're not getting stronger you're getting weaker.
Now, here's the key. If you want to be solid at 5.11, don't look at where you need to be to climb 5.11, look at where you need to be to climb 5.12. If you aim to become that person, 5.11 will come easily. Same with any of these grades. If you want to be solid at 5.x, aim to be a 5.x+1 climber. Train like that climber, eat like that climber, sleep like that climber. The rest takes care of itself.