You need Rock smash to get through Rustboro tunnel and route 111, Strength to get through victory road, Surf to get across many routes, Waterfall to get to the league, and Dive to get to Sootopolis. Fly is completely optional, as usual, but you won’t have access to the soaring feature until after the 8th gym. Cut also isn’t needed to progress in the game. It only opens a few new areas.
Don’t keep rock smash or cut through the entire game. You only need them a few times at the beginning of the game. Teach those to a slave and keep it in the pc. Latias and Swampert can learn all the HMs you’ll need, if you use those. Sharpedo is another good one if you don’t have a water type.