Should I let Pikachu become a Raichu in Pokemon Blue?
Yes. Also just find an HM slave pokemon that's actually worthy of being an HM Slave. Most normal type poke can learn Cut, Strength, and Surf. I know the amount is more restricted in RBY, but they are around. No need to completely restart and then waste a Mew with being an HM slave lol.
Also in Gen1, Flash is useless in my opinion. The only place it helps is in the Dark Cave, and you don't even actually need it for that, so ¯\_(ツ)_/¯
HMs are in every game btw, and as the gens go higher(more recent Pokemons) they added in new ones to be even more annoying(Waterfall, Dive, Rock Smash etc.). And oh yeah, btw, in Gen 2 when you get there, Rock Smash is pretty useful but is cotained in a TM, instead of HM, so you can screw yourself out of using it for its intended purpose.