Well, you're describing exactly how to do it with bind9 :) What make you say Unbound is better?

Unbound is more lightweight. All it does is recursive lookups, it has no option for being authoritative. If that's all you need then that's better.

There are other options too, I don't know what makes Unbound better than PowerDNS Recursor or Knot Resolver.

I prefer Unbound as it lets me set a min-ttl. This of course is taboo and comes with caveats, so you have to know what you are doing and the implications. Unbound has better support for DNSSEC as well.

Unbound is a bit less likely to come with distro-specific oddities in the default config, I think, mostly. It's a matter of preference.