I've read Louie Simmons say that the deadlift is not about the forearm strength, it's about training your fingers to hold on to the bar. In that respect, supporting strength is bound to fail, and then your fingers are the last to hold on and they have much greater capacity to maintain grip than your forearms staying closed. So training finger strength would be wise, as well as supporting strength.
In general, I don't train pinching strength because it has no carryover to my lifts with an axle bar, but the axle bar increases thumb strength (albeit more slowly) and increases my plate pinch lift. Crushing, however, should be trained through sandbags or hand crushers. I train supporting strength once a week (farmers walk, 1-arm holds with bodyweight on a powerrack, farmers walk holds). I train with 2" bars for assistance work + deadlifts, and I work in crushing grip whenever I have 2-3 days where I won't be using my grip excessively. I don't try to "train" it like a deadlift or anything, I just make sure I use all facets of grip and make sure I'm advancing from workout to workout through reps or weight. Given there's so much to grip strength, when I've tried to systematically record it all and progress, it just got to be too much. As long as you're working in all facets of grip (pinch, supporting, crush, wrist, fingers), you'll have all your bases covered.
Some good lifts to try out and advance on: axle deads (and dead variants), axle bar rows/curls, kroc rows w/ 2" DB, axle press/clean, farmers walk with towels wrapped around farmers bar handle or just plain plates, axle bar chinups, 2-finger chinups, chinups on ledges/rocks.