Ah yes, text dump time. Please disregard immediately if an originals fan, you'll find naught of interest.
No, to me it wouldn't be that bad if it wasn't a Fallout game. You're not going to find much opinions like mines here because this is a forum for cRPG fans and without the association with Fallout, an open-world FPS would be of little interest to everyone here. It would probably get two or three mentions on that "What game are you playing right now?" posts, and nothing more. Keep in mind that I'm not a devout fan of cRPGs and therefore will not have the popular opinion here.
From the
least biased perspective possible, I guess it would garner moderate interest with its integration of settlement building into sandbox gunplay, but its lack of co-op or any multiplayer means it would fall out of popularity and relevance as fast as Rage did. The competition, in this case the Borderlands series and all its clones, only manages to keep the repetitive design entertaining because of its humour and its multiplayer (which gives a game a very long shelf life).
If this theoretical exactly-like-Fallout-4-but-not-Fallout-4 shooter would be identical in every way to Fallout 4, then the lack of good writing or any multiplayer feature would kill the game pretty quickly. Make no mistake - it's not a bad game at the core, but it sold because of its marketing, not because of innovation. The massive force behind the Fallout brand and plastering it everywhere after Fallout 3's success mirrors the path Activision took with Call of Duty.
Then we step into modding territory, which is harder to discuss, because it's very dependant on who exactly is handling the game. If this theoretical game handles modding the same way we've seen Fallout 4 handle modding so far, it would be dead in the water. If it was handled by a competent developer that cares for the fanbase, then I suppose the modding would keep the game afloat for much longer, but like I said, no co-op and no extreme marketing campaign means that it wouldn't be nearly as big of a phenomenon as Fallout 4 currently is. Maybe replace "phenomenon" with "clusterfuck".
So in summary, it would be decent, but practically unheard of. If anything, the character creator, settlement building and moddability (if that's a word) combined could possibly keep it interesting, but frankly I doubt it. Anything its competitors haven't done, MMOs and those DayZ survival clones that litter Steam Early Access did it long better. Mods would be the game's sole trump card. If it's played wrong, it will be treated just as Rage and Dead Island did - above average, good future potential, but brought down by its flaws.
Mods and marketing keeps Bethesda up high. If at any point they drop the ball on either of that, well...