Now for this character there is no screwing about with potions and books and arcane gestures. He is simply a force of supernatural rage and badassery, whose abilities are throttled by the same chart that limits other supernatural buttkickers. Would this be an acceptable character? If not, is there an objection that does not lie at the metagame level of labeling his abilities as spells?
It's definitely getting closer.
Now we need to take one step further and make these kind of 'spells' not dailies, to make it feel physical rather than abstract/intellectual (and to differentiate from casters too, as we don't want all classes to play the same). Let's have a stamina system for instance. But look, we have it already: we can use the Battle Master Superiority dice for that!
And I like the use of concentration you suggest.