Star Trek (2009) (and later Discovery) established that alterations in the timeline do not necessarily mean the original ceases to exist, since Prime Spock and the Prime Universe still co-exist even though the Kelvin Universe branches off with Nero's appearance in the past and the USS Kelvin's destruction. And the changes are a matter of perspective for the people caught in the weirdness.
Whether time travel causes a branch or alteration of the original timeline is all over the place in Star Trek. For this episode, my guess is that La'an could only use the time-travel device to return to her timeline since the device comes from a Department of Temporal Investigations that's native to the future of her timeline.
Within the context of Star Trek, from
1966 to 2005, they didn't do the branching timeline thing. They had the timeline altered by changes made to the past (City on the Edge, Yesterday's Enterprise, First Contact, Past Tense, Future's End, etc).
JJ's Reboot brought on the new universe branching concept, mainly as a conceit to have a reboot while not alienating the existing fans.
Although, if you follow this line of thought, does first contact with the Vulcans get pushed beyond 2063 too?
Supposedly a quote from Akiva Goldsman
It's what I mentioned earlier. They will probably push off WW3 as well, it would go against the messaging of the Nu Trek creatives to have a decent to barbarism, bloodshed, anti-wokeness, and slaughter so close to the modern day and the
progressive world that they want to see.
They're having trouble presenting Star Trek as an alternate timeline to our real world.