Bush was able to "accomplish" those wars because the country was "united" after 9/11. It didn't have much to do with his personal competence or connections.

The US government (like all liberal, democratic governments) is a massive, many-headed bureaucracy, with competing and sometimes contradictory objectives. Those objectives are long-term, shaped by committee and subject to organisational inertia, so the departments will function regardless of the political weather. It doesn't matter what public sentiment is, nor how widespread that sentiment may be. Being able to utilise any part of it requires connections and a high level of managerial competence - in George W Bush's case that was through his own personal, familial connections, and through his cabinet, many of whom were career civil servants who'd served in previous administrations going back decades. The feeling of "unity" people may have experienced doesn't come into it.

That certainly helped, but it was well known even then that Cheney had a lot of power, and he had many connections. GWB wasn't a complete puppet, but he wasn't fully running the show either.