I am not sure they want an actual arm race, because I am sure they haven't forgotten how Reagan brought them down economically with that in the 80'ies.
What I am sure that they do want is that NATO and friends will back off and not do anything, as we have done in so many places, Ukraine and Syria to name a few famous ones.
Every time Putin rattles the nuclear saber Western and NATO countries rattle their knees, so Putin has no reason to have _any_ respect for whatever we _say_.
A full scale war is not even necessary, and also something we don't want, but shooting down a few Russian planes and helicopters and destroying a few tanks and other important equipment that can easily be seen on media, so the Russian public can also see it, _then_ Putin will listen. That is the only language he understands, being a USSR hardliner.
Also, USA should not _excuse_ killing a number of Syrian Al Assad forces by saying it was a mistake. I hope it wasn't a mistake, and it should be strongly emphasized that there is more where that came from, unless _they_ back off!
Russia and Al Assad doesn't want any civilian to stay in Aleppo, as this is a genocide if there ever was one!
NATO and liberal countries in the World, wake up!