Well he certainly made a big fuss the first time around, but it didn't stop him from losing. I can't predict the future but I also struggle to imagine how the "third term" scenario would even go. Trump's reelection issue is persistently that his policy-making is so poor that it basically advertises the opposition as soon as the primary starts. The DNC just has to pull back their radical candidates and push an affable nobody like Joe Biden and collect their free win.

The GOP's support isn't guaranteed throughout this administration, and they know a successor needs to be found. Amending the constitutional term limit would be a source of bipartisan outcry, because neither party would survive it.