You DO NOT need to be loose-limbed. You need to hold your body tight when tumbling. If you are loose you are going to loose power and slow down. You want a strong, tight body when you are tumbling. If you don't have good body tension, especially in your core, that could be an issue.
Another possibility would be if you are bending your arms in flics and handsprings. By bending your arms you dissipate the energy of the impact. It is like when you bend your knees to stick your landing. That dissipates the energy and helps you stop. If you want to rebound you need straight legs or arms. You shrug your shoulders to block when coming of your hands, which is the equivalent of punching with your feet, when your legs are straight and you use the flexibility of your feet and ankles to quickly rebound.
There could also be issues of technique. In a handspring or flyspring for example a common error is to reach down towards the floor in the hurdle which closes the shoulder angle and causes a sub optimum angle into the skill. Another common problem can be hurdling too high rather long.
There are a number of different reasons that could cause your tumbling to be slow. Try to pin your coach down on the cause. They are supposed to tell you how to fix it. It may also help to identify problems if you video your tumbling and watch it back, preferably at low speed.
If you're wrists get sore you may need to look into getting tiger paws/ wrist guards: http://www.tigerpawwristsupports.com/tig...
As for powerful tumbling, the run is important but you also need a powerful hurdle into the skills.
Also make sure you push through your shoulders if you are doing handsprings.
And make sure you don't come in to high on your skills.
I think you might need to be loose-limbed, if you know what I mean? Think of dancers whose bodies flow like water. Yoga can help with that, a steady practice. Given that you're athletic, maybe the one with extreme heat Birka(?) yoga.
tricky issue try searching using a search engine that will might help
