Has avoiding Cloudflare become Impossible?
Mostly, yes. But let’s break this down. Cloudflare only breaks web services and so far Cloudflare’s privacy abuses and gate-keeping is mostly confined to the web. Avoiding Cloudflare is impossible in some circumstances.
CFd government sites are unavoidable (voting rights lost in the US)
The only Cloudflare sites that are strictly unavoidable AFAICT are government sites. You can always boycott the private sector, but the public sector is shoved down our throats. There are 6 or so states in the US where voter registration goes through Cloudflare. Even if you register on paper there is still no escape because the data entry worker likely uses the Cloudflare site. I am a non-voter for this reason. Although it’s still possible to move to one of the 44 other states and register there.
CFd medical websites
See How lack of digital rights, Cloudflare, and Google worsened a medical emergency situation and undermined human rights. When you need medical info in a hurry, boycotting is tough.
search is liberated -- but only by 1 single search service to date
There is only one general purpose search service that helps avoid Cloudflare: Ombrelo, which tags and down-ranks Cloudflare websites in the results.