Seeing as Basculegion seems to be this game's answer to Surf, Hisui Braviary replaces Fly, and Wyrdeer seems to be filling a similar role to the Bike in past games (like Ride Tauros in Alola), I can imagine a theme where every new regional variant or regional evolution in Hisui is some kind of stand-in for a HM or other old mechanics. Those leaks from a while ago that show what looks like a monkey Pokemon scaling a hill could be this game's version of Rock Climb, and then we could also have a Rock Smash one, a Strength one (Hisui Arcanine?), and perhaps even a Dowsing Machine one. I won't say for certain this is what they'll do, but it looks like that could be the theme they're going with.
Like the idea of a Dowsing Machine ride and that leak with the climbing mon (Nuzleaf split evo?) is basically confirmed. Remaining HMs/Ride functions (from any game) besides Surf and Fly are as follows:
Cut, Defog, Dive, Flash, Rock Climb, Rock Smash, Strength, Waterfall, Whirlpool, Stoutland Search, Mudsdale Gallop. We can probably safely remove Cut (no ride equivalent in Gen VII), Defog (exclusive to Gen IV, even with fog being brought back in Gen VIII), and Whirlpool (little usage and later phased out). This leaves Dive, Flash, Rock Climb (leak image), Rock Smash, Strength, searching, and crossing rocky terrain. My personal guesses are that we still have yet to see Rock Smash, Strength, a searching mon, and maybe Dive to look forward to. Maybe a Primeape evolution, your Hisuian Arcanine idea, Hisuian Donphan (both Stoutland and Donphan evolve from Pokémon with Pickup as a possible ability), and a Hisuian Wailmer?