The GM can only do so much as far as convincing ownership of what to do. Personally, I think Henry and Ben were very much on the same page. And I think that their strategy of not handing out the big contract to an aging pitcher was the right one, though I think they could have re-signed Lester for a reasonable amount had they done it during ST.
I don't think Dombrowski had to convince Henry of anything. Henry, along with this fan base, just didn't have the patience. Signing Price was a knee-jerk reaction to the last two seasons. As I just posted, not his wisest move.
It doesn't sound like Dombrowski convinced Ilitch of much when he was in Detroit. In fact, it sounds like Dombrowski was more or less a puppet, doing what Ilitch instructed him to do.