Star Trek and Stargate both go by the premise that “anything that can happen does happen in alternate quantum realities.” It’s just that the characters from those realities assert that their version of reality is the only one that matters (which is an actual quote from Teal’c.)
The Prime Universe does still exist. As does the universe in Yesterday’s Enterprise and the Mirror Universe. To say that Prime does not exist is to say Mirror does not exist; and the Mirror Universe’s timeline, which is wholly separate from Prime, was accessed repeatedly by Prime characters (and vice versa) during the events of Deep Space Nine, not to mention TOS and Enterprise.
Star Trek 2009 nine makes it clear that the Narada when back in time, altered history and thereby turned the prime universe into the Kelvin universe. There may be an alternate universe that is exactly the same at the prime universe, but it would only be a duplicate universe and not the prime universe. As far as I know, Discovery does not take place in an alternate universe that is a duplicate of the prime universe, it takes place in the prime universe itself.
It actually specifically makes the opposite clear. The Narada created a divergent timeline that they call “an alternate reality”. The Prime universe is still there, untouched (just without Romulus). See also: Star Trek Online, which continues in a post-Romulus Prime timeline.
No where in Star Trek 2009 does it say that the Narada went to a different universe, it just went back in time and changed history.
That’s not how time travel in Star Trek works. You should watch the TNG episode “Parallels” again. In that episode, Troi was not worried about being non-existent should Worf fix the space-time continuum; she was worried that her version of Worf, to whom she was married and had children, would not return to be with her after all was said and done.
The timelines that Worf encountered in that episode were “alternate timelines” that existed before that episode and continue to exist afterward. The Trek multiverse does not have a single line of causality in which only one version exists at any one time. I’m certain we do, here, in real life, but Star Trek is fiction. So there doesn’t have to be a real life that-doesn’t-make-sense-from-what-I-know-of-God reasoning for why Trek must have a single timeline.