Imagine the OTA weighing in on the recent net neutrality shitshow. Wouldn't have made the FCC look to smart would it have? That same threat existed for every agency and congress and the executive. Literally everyone wanted them gone.
If anything, a fact checking organisation has incentives to align with or even become a lobby group. We've seen what happens to "fact checking" in the political sphere.
Some would say, if a senator doesn't believe in climate change that's a question of fact, and if they merely received better advice they'd change their opinion.
A cynical person, on the other hand, would say they know full well that climate change is real but they've accepted money to pretend they believe otherwise. In that case, no amount of independent advice will change their minds, as it is not a question of facts but of money.
No, no it shouldn't
There's a really, really, really fine line to walk between preventing the AMA from lobbying congress to screw us and preventing actual grass roots people from paying someone to argue on their behalf.
Besides, without lobbying groups working against you, the feedback loop of voting them out actually starts working. (no it's not a complete solution so please don't argue against that straw man, it's just the most important first step to unblock others)
Edit: clarity
This should be step one, always.
While those perverse incentives exist everything else is an uphill battle.