> they do have a large amount of influence
Put another way, they are allowed to act unchecked by Congress much of the time.
Why is this?
I think the president and congress play a coordination game through which corrupt legislation is passed.
Take Obamacare. Its passage was a massive success to rent-seeking firms and interest groups. All the interest groups and most of the members of congress who were instrumental in getting it passed were happy to have it called "Obamacare" and not "PhizerCare" or "AbbotCare" or "HalliburtonCare".
Correspondingly, both parties benefit from the notion that Obamacare is somehow leftist or constitutes a big defeat for "greedy firms" and "selfish doctors".
In reality, the program simply modernizes many of the existing flows of rents from the public to firms, and reflects some of the changes in magnitude of influence of some industry subgroups.
This is one reason why the GOP didn't come up with a serious market-based counter-proposal: The spoils of Obamacare were just too appealing to pass up, at least for enough of the GOP's constituents to get the bill passed. The loud crowing about wanting to repeal it is no more meaningful than anything else politicians say to seem principled or to seem consistent.
From the beginning, Obama was far more pragmatic than HRC about pleasing healthcare industry incumbents, which is one reason for his surprising performance in the 2008 Democratic primary.
I strongly agree that executive orders and extralegal actions (which Bush honed and Obama perfected) are quite worrisome... But the core issue is that congress largely agrees with the actions that are undertaken. As a case in point, there was a lot of outrage from Democrats about GWB's extralegal shenanigans during the Iraq war, but then once Obama took office we got the chance to observe that in fact he (and his party) strongly support drone warfare, happily exploit the idea of enemy combatants not having rights, enjoy every clause of the Patriot Act, and utilize every extrajudicial rendition option just as GWB did.
We have one party, and it goes through a little ceremony every few years to see which corporatist warmonger gets the chance to be our figurehead.