To check the DNSSEC signatures on the client, you have to do a full recursive lookup. You've always been able to run your own DNS cache, if you want your host to operate independently of any upstream DNS server. But at that point, you're simply running your own DNS server.
It's not necessarily equivalent to a recursive lookup, you can ask a cache for all the answers because you already know the root keys a priori. But yes, it does follow the entire chain of trust, that's the entire point of dnssec: if you don't do that the whole exercise is utterly pointless.
It's explicitly not the point of DNSSEC, which has for most of its entire existence been designed to be run as a server-to-server protocol, with stub resolvers trusting their upstream DNS servers.
Not true, RFC4035 says all security aware resolvers SHOULD verify the signatures. It's far from pointless when actually implemented. Don't dismiss a whole protocol just because some historical implementations have been half assed.
I'm guessing I do. Anyways: no question that there are a variety of experimental setups in which you can address the problem of on-path attackers trivially disabling DNSSEC, freeing you up to work on the next, harder set of DNSSEC security and operational problems.