That's very true, and there are similarities in the awareness that they develop, but it isn't very meaningful to call e.g. judo 'thinly veiled mindfulness meditation' because the practices are so different. It's the same with Feldenkrais, where you're doing a specific repertoire of exercises that aren't much like these other things.

The Feldenkrais exercises have almost an algebraic quality (or perhaps I should say combinatorial) in the elaborate way they use symmetry. For example: do something on side A, then do it on side B, then vary it by X on side A, then vary it by Y on side B, then Y on A, then X on B, then do those at the same time, then switch and do X on A and Y on B at the same time. As you go through these permutations you can feel your brain noticing symmetries and contrasts that it wasn't aware of before. Sometimes you become aware that a simple movement is possible which your body had no idea it could do. For example, that you can move your arm behind your shoulder in a way that you would have sworn was impossible, or at least painful, and yet it's totally easy. The method seems to work by rewiring your mental model of your own body, mostly by expanding your awareness of what it can easily do. This is related to pain relief.

One of the strangest effects is that it works better the less you do. A common line the trainers use is "however much you're doing, do half as much". This is probably because the method works by getting the brain to notice finer and finer contrasts. But it's surprisingly difficult to do less, because in most aspects of life we're so habituated to thinking more is better.