That's not the case, though. You might be somewhat disadvantaged, but not seriously.
On my first playthrough on normal difficulty I essentially only used the Super Shotgun. I can't remember using the Chainsaw for ammo, but I did use the BFG a handful of times.
The developers have even said that a goal of theirs was to make this "playstyle" (using only the SS) less viable in Doom Eternal, because it was too simple and powerful. It might have been on a NoClip podcast, I can't remember OTOH.
I've not played Eternal yet, I'm waiting for updated nvidia drivers to hit my distributions repositories.