It's a similar process to Cloudflare. Point the NS to them and enable the proxy for a domain or subdomain.
(don't use CNAME flattening with DNS-routed CDNs like Bunny though, if you must use an apex domain then use the CDNs integrated nameservers)
Has this situation changed?
Bunny has a similarity concept: https://bunny.net/edge-scripting/
What is the problem with doing that?
I've been trying out Bunny recently and it looks like a very viable replacement for most things I currently do with Cloudflare. This new database fills one of the major gaps.
Their edge scripting is based on Deno, and I think is pretty comparable to e.g. Vercel. They also have "magic containers", comparable to AWS ECS but (I think) much more convenient. It sounds from the docs like they run containers close to the edge, but I don't know if it's comparable to e.g. Lambda@Edge.