You shouldn't need the HM to beat the game, as the only thing it is necesary for is teaching Fly. (there are other Flying-Type options for moves) Fly only has use out of battle as a convenience factor, so it shouldn't be needed to beat the game.
In addition, there is also the eon flute you can obtain later in the game which removes the need for fly.
The only thing you loose out on is opening the chamber for Registeel (which can be worked around by evolving a Bagon into Salamence and learning Fly through level-up)
In short: you should still be able to complete the game without thinking too much of it.