Earlier quoted context omitted.
The problem is that's a false narrative. There were never any promises, and Putin barely even cared that it resulted in Finland and Sweden joining NATO. Because Russia's stated motivation for invading Ukraine was never Putin's actual reason, which is basically an emotional desire for historical greatness by reclaiming Russia's lost empire, combined with war always being an excellent mechanism for staying in power and…
But how do you know so clearly that the one thing is a lie and the other is the truth? Even if you reject the "encroaching NATO" narrative, what makes "Putin just woke up one day and decided that remaking the Soviet Union and/or the zarist Russian Empire would be a great thing to do in the 21th century" the more plausible hypothesis? What information do you have that Mearsheimer doesn't?
As to a more realistic narrative -- it's a bit more nuanced than the formulation you suggest, but even so -- is pretty much obvious once we look at the things Putin and key people around him have been saying, along with the last 350 years of so of Russia's history vis-a-vis its neighbors and Belarus and Ukraine in particular.
The links in the short comment tree below (which answered essentially the same question from just a few weeks ago) might be useful here: