Virus premise doesn't gel

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that's the thing you worry about? probably the most plausible thing in this movie, I very much so like. Just rewatched it after so many years/lustrums.
It's a well-worrying and common thing in animal agriculture. That's why the Wuhan wet-market wasn't as implausible at first cause these wet-markets supposedly sell any meat they find in the bush. In the west, when you hunt game, you must send meat probes to a lab before being allowed to process/consume it.
https://en.wikipedia.org/wiki/Zoonosis
"A zoonosis (plural zoonoses, or zoonotic diseases) is an infectious disease caused by a pathogen (an infectious agent, such as a bacterium, virus, parasite or prion) that has jumped from an animal (usually a vertebrate) to a human.[1][2][3] Typically, the first infected human transmits the infectious agent to at least one other human, who, in turn, infects others."
Also one of the main arguments vegans use to wean off people off a meat diet.

It's a shame they "ruined" the movie with the premise. Then again this was the 90s, shit like that didn't matter. God I miss that decade.

In the future, they're able to invent time travel, but they can't treat/cure/vaccinate against a virus.

Maybe it's because there are a bunch of future people still going outside without HAZMAT suits on going, "It's all a bunch of fake news by our time-travelling overlords! There's no 12 Monkeys virus! Psh! I'm goin' outside!"

I can try to suggest a sociological reason for them staying underground. In modern life, we live with a false sense that technology and progress protect us from any disaster that, in the past, would have cost thousands or millions of victims. Death is not a daily presence in the 1990s as it was during the Middle Ages. In those times, corpses in the streets were a common sight, even when there was no plague. Modern man is unable to cope with such horror. In the movie, 5 billion people killed in one year, almost 99% of the world's population in 1997 (!!) - it's much worse than the Black Death.

We are talking about collective anxiety and cultural anxiety that is not likely to end in one generation. And this is even before we start dealing with the loss of millions of professional and expert personnel who could help society recover faster. On a psychological level, they would rather live underground at this stage, with all the lack of freedom and difficulties, than take the risk and return to the disaster area. The entire society is dealing with anxieties and PTSD. Regardless of the state of the virus as of 2035, it will take time for them to gather courage and return from quarantine.

It's funny you chose that as unbelievable and not the time travel part.