A backlash, certainly by those that are well informed will happen  As for the general public that is less informed and a bit lazy/naive - it will depend on how is presented.   The government giving out CBDCs and being presented as way to:

1.  protect the US dollar against foreign adversaries
2. combat fraud and crime
3. Give tax incentives
4.  Insure against bank failures

etc...  when presented in that way and possibility with incentives, much of the general public will fall for it and not see any alternative.   There will also likely be the various political presentations to appeal to both the left and the right.   The problem is more than just CBDC vs BTC & Cryptos - the problem is the public is lazy and will take whatever path requires the least of them and appears to give them the most.

There will most certainly be a battle and I suspect it will last a very long time.

if you look at what happened in the USa in 1971 when Nixon took the dollar off the gold standard and go with full blown fiat, the pay off was making it legal for USA citizens own gold again, people just saw the idea to own gold a good pay off, again make forgot about how the government stole the gold from everyone back in the 1940s

The mainstream media does tell everyone that crypto is a scam and many believe that, so more less