If polling were to reveal a majority of either party were more open to nuclear strikes than their predecessors, that gives policy makers a signal and an opening.
Deescalation stopped because of people in general not caring enough (and making money of being the biggest power), not because of administrations that come and go.
As to the immigration situation: we know that governments are not executing in general how they should be, but people are able to enforce some policies if they fight together united and in agreement. But right now they are not in agreement.
There was only one administration with that opportunity, really; Truman.
Every other administration has had a nuclear armed Russia in play.
Attempts to do what you describe were still quite common, starting as early as the 1950s. https://en.wikipedia.org/wiki/Nuclear_arms_race#Treaties
55% of Republicans say ICE's efforts are about right; 23% think they don't go far enough [1]. There is limited evidence Trump has lost touch with his supporters on this issue. The question is if this is this GOP's pronoun issue–popular in the base but toxic more broadly.
[1] https://www.ipsos.com/en-us/where-americans-stand-immigratio...