Well I guess there is a chance he personally would have survived had he done so, but it seems unlikely. Had he stayed in Nevrast, his people would have been involved in the
Dagor Bragollach, I imagine some going up to to defend Hithlum and others going to hold the pass of Sirion and aid Dorthonion. We do not know many details of the Bragollach... such as how much time went by between the various attacks, so it may be hard to say exactly what the
elves of Nervast would have done. I think the involvement of Nervast in Bragollach would have made a huge difference in the outcome of that battle. Hithlum was held and Dorthonion and Easterward Marches lost.
Fingolfin and
Fingon were cut off, unable to reach Dorthonion to aid there. Had
Turgon's people been active, the elves of Hithlum may not have been cut off from Dothonion at all, and their aid (or that of Turgon and his people - likely both) might have been enough to have victory at Dorthonion, and perhaps then
Maedhros would have ended up with even more aid. It's hard to say... we also don't have numbers. I guess the siege would have still been broken, but there is a chance a lot of Turgon's people would have survived, and especially if they stayed at the pass of Sirion or went up to defend Hithlum versus going to Dorthonion and being wiped out. Now if this stuff had happened, there would probably not have been an oath to Barahir, as
Finrod may not have become cut off. But that is beside the point

.
Assuming Turgon and enough of his people survived to continue the
realm of Nevrast, and assume they chose to do so (they may have gone elsewhere now that the siege had been broken... such as to
Nargothrond, Hithlum, or scattered to various
realms) then they would have been at the Nirnaeth, and Turgon would have probably fallen there. The only reason Turgon escaped Nirnaeth is that he had that
Hidden realm. I do believe he would have fought to the death if not for that.
So, Turgon would probably not have been better off... unless you want to look at it this way: He probably would have felt better dying in Nirnaeth with his brother, than dying knowing his refusal out of pride to ignore the last counsel of
Ulmo had been a death trap for his people. But even then, one is assuming a lot. We can't be sure how Turgon would rather have died.
And even if somehow Turgon did get out of Nirnaeth alive, I suspect he might have died by other means... in Nargothrond, for example.
That is in answer to the question you ask in your post. As for the poll question 'Should he have stayed in Vinyamar?', I vote no.
Ulmo counselled him to go to Tumladen, so I believe this was the best course of action. The only way the answer could be 'yes' is if one believes that Ulmo who 'appeared' to work against the Doom, was actually part of what brought about so much ruin for the
Noldor, and that while his plan allowed for the voyage of
Earendil it was also the cause of much ruin, and the Noldor woud have been more successful without following his counsel.