Cloudflare will provide you with certificates they generate, that they verify but won't be accepted by anyone else. (No cost because of that) - this keeps the data secure between you are them. Obviously, you are still trusitng cloudflare in the middle, but still less trust required.
If you can install a certificate, you can already get a real one from Lets Encrypt (you don't actually need to run their client on the server). The problem is that many shared hosting services are still stuck in the past, and don't let you use SSL/TLS at all.
Without running the client, that means manually changing the cert for expiry, which is very short on LetsEncrypt certs. That introudces the possibility of forgetting or messing it up.
I agree that the best option is for shared hosts just to build in support for LetsEncrypt.
Right, so I've currently got CloudFront in front of it, but doesn't that move the problem? Now the connection between CloudFront and S3 is unencrypted.
(I'm probably understanding this wrong, but I'd like to understand why.)