Surely misinformation has had an impact, but how big?

I wonder how that compares to the perceived incompetence.

Some official sources are wrong some of the time, everyone's favorite example: they told mask were useless in the beginning. They corrected course, but then told vaccinated people they didn't need masks anymore, basically making the same mistake twice. They halted AZ, paused J&J, had a gazillion change of mind on boosters. We need to be able to hear challenges to official sources.

So yes, the chilling effect can make it worse if they go unchallenged. And it's also not simply true that free speech would return either.