So moving this away from a “what constitutes an h shape,” and back to the original question, I think outside of 5A there’s no significant benefit to an h shape over v, w, or bell.
If you look at the Shutter compared to the Genesis (which to me seems like a heavy influence on the Shutter), one of the big changes is reducing the flare of the rim and angling it down towards the gap to make more of a v shape, widening the catch zone. Which makes complete sense.
But that doesn’t matter as much in 5A and I think having that h shape lends itself well to 5A needs like stability and keeping the string moving in the direction you want while it’s being manipulated by the momentum of the counterweight.