Concurring with you, and here's some entertaining food for thought. It seems to me that every virus must have a minimum host population that it needs to persist.
The way I think about it is like a tabletop game. You, the player, are a virus. You draw a strain card and roll a stat sheet. Then you pick a human to infect. They draw an immunity card. After that you roll a set of dice that determine if you get to "evolve". If you evolve you get to draw a new strain card, replacing your old strain card, roll new stats, and all players discard their immunity cards. Whether you evolve or not, you start a new round by picking a new human to infect and repeat. The big caveat: you can't pick a human with an immunity card.
Now, if the number of players in this game is low enough and the likelihood of you rolling an evolve action is low enough, then it's easy to see that in all probability you'll lose the game. Every human will get an immunity card before you roll an evolve.
So there are two variables: host population and likelihood of evolution.
Our goal with vaccination is to decrease the former as much as possible. If we get it low enough, the virus will lose the "game". Every other non-vaccinated human will gain herd immunity naturally before it has the chance to evolve. It will die out, and never be given a chance again.
Now, I'm not addressing herd immunity. That's about whether a virus can spread in a given population. This tabletop game doesn't incorporate a virulence mechanic. This is about whether a virus can mutate before the _globe_ achieves herd immunity. The point I'm addressing is this idea lurking in the back of people's minds: third world countries. Won't the virus just "fester" in countries with low vaccination rates until it mutants enough that it can become a new pandemic?
My suggestion is that it's not a given that that's the case, as long as whatever remaining non-immune population is small enough.
What is low enough? An actual virologist could probably guesstimate for some given probability threshold.
But the good news is that SARS-CoV-2 has really bad evolution stats. Many viruses have a "checksum" protein in their genome, just like most other organisms do, that actually work to prevent mutations. Some viruses have this protein "tuned" lower so that they mutate faster, but it's a trade off because that often results in more production of impotent viral particles. SARS-CoV-2, from what I've read in studies, has its "checksum" protein tweaked higher, so it just doesn't evolve as quickly as something like the common cold strains.
In other words, I completely agree with your point. Get vaccinated as quickly as possible and keep up masks and social distance for now. That will give us the highest possibility of winning this "game".