Civil War didn't really explore that angle that much beyond that one mother that guilt tripped Tony and some valid concerns raised about their mission with the then (or current) inpexperience Wanda on the field. Then it just became about Tony not liking that Steve's ex showed back up and the framing of Bucky. I'm sure there might be some blame game story post Endgame but I doubt it will be spun as the Avengers are a bad idea to the public. Just that the goverment is not cool with them if they can't control them.
I think it's possible to flash back the origin of the X-Men and the first few years as a quick reel in the movie. Almost like the opening credits of the second hulk movie which jumped through the origin story as part of the opening credits. If they decide to reflect the current comics we could go back to the old theory that all of the X-telepaths wiped humanities memory of mutants so they could be safe. The Sokovia Accords are very similar to the mutant registration act so there is definitely a pre-existing sentiment in the MCU to persecute anyone born with powers.
There was also the news report after the botched mission where the public blamed the Avengers for the deaths that were caused. The premise of the plot resolved around someone hating and blaming the Avengers for a tragic loss. The film put enough out there to let us know that collateral damage, though rarely touched, is a real thing in this universe and those affected hate the Avengers for it. The MCU tended to sugarcoat and ignore that until CW but it does shape how they are perceived