I agree that this exposes the risk of relying overmuch on handful of large, opaque, unaccountable companies. And as long as Cloudflare's customers are web operators (rather than users), there isn't a lot of incentive for them to be concerned about the user if their customers aren't.
One idea might be to approach web site operators who use Cloudflare and whose sites trigger these captchas more than you'd like. Explain the situation to the web site operator. If the web site operator cares enough about you, they might complain to Cloudflare. And if not, well, you have your answer.