No I didn't want to use an external filter since FAST is meant to index PDF out of the box. Using a 3rd party filter would lead to performance issues.
Nevertheless, I found out what was the glitch. The Account used for the connector MUST be the same used for stating FAST services. Don't ask me why (if you use docpush on the FAST server, you don't even use sharepoint and the push is denied by FAST.
Anyway, now, the crawl and the index work like a charm.