Only because those radicals have thus far been useful idiots. Trump certainly bit them in the ass and turned the party on its head, but the party is still supporting capital. Had he come out the gate with a crusade on corporate America his campaign would have faltered very quickly.
There are a couple definitions. One I’ve heard most is a republic has a citizen as head of state, which disqualifies both monarchies and military dictatorships. Another is that the head of state is elected or nominated, which disqualifies non-representative systems entirely.