Of course that's the main reason, but I think the concern in this thread is that why is it RUN, not BEGIN, GO, PLAY or other similar words that consumers (at that time) would more likely understand. The use of I and II (or 1 and 2) is obvious and logical, but not RUN, apparently. It is quite possible that as NEC itself was a major computer manufacturer (not to mention that Hudson itself wrote implementations of BASIC on various 8-bit computers, that included Family Basic on the Famicom; too bad its sorta equivalent on the PCE, the Tsushin Booster, got canned), so they're in the mindset of "running" or "executing" programmes, and indeed it was a time where BASIC was a prominent programming language on home computers, so there were at least people who were familiar with "RUNning" their BASIC programmes.
Anyway, this (alongside the not-so-fortunate truth of having only one controller port) became one of the most well-known characteristics on the system.
"Run" *is* a similar word that consumers understood at the time. In that context it seems a little Engrish but no more so than "Enter/Return," or "Escape" on a PC keyboard.
When I "run" my car I kinda have to "Start" it first, right?
When I "run" to the store I kinda have to "Start" my trip, right?
When I "run" a story in a newspaper, I kinda have to "Start" publishing it, right?
Wen I "run" a campaign, I kinda have to start the campaign, right?
When I "run" a game, I kinda have to "Start" the game. Same thing. "Start" is understood to be "Start and Run." "Run" is also understood to be "Start and Run."
If anything, "Select" is more Engrish than "Run," since it changes your selection rather than making/confirming your selection. IOW, Select does not "Select" your selection. Run, on the other hand, does "Run" your selection.