So why do companies pay lobby groups?

To the simpler minded among us business lobbies against regulations because such regulations would hobble business. And of course those who would impose regulation upon business are the very driven snow in purity, seeking as they are to restrain the depredations of business upon the citizenry. However, it doesn't really work that way:

But the fact that corporations also fund big-government organizations raises questions about this narrative. If regulation hurts corporations, why are they funding think tanks which promote it? The truth is that most regulation is written by and for incumbent businesses to erect barriers to entry and to buy advantages over their competitors.

I'm afraid that it's true. Big business just loves regulation for it prevents those pesky upstarts from muscling in on their profitable territory. Examples abound: Uber is prevented from offering an electronic method of hailing a cab because this would affect the incomes of the incumbent cab owners. The regulations on copper smelters in the US are such that it would be impossible to build a new one and actually meet them. All of the extant smelters are of course grandfathered in to those same regulations. There can be no future competition. Here in the EU it takes 10 months just to get planning permission to set up a new industrial production line. And yes, they have just made the rules stricter, this applies to any sized production of anything.

Big companies don't mind this as they have the time to fill in the paperwork: and the €10,000 to file it. A new company, an upstart, would find it most difficult to stay alive during a process of such length. Which is why the big companies just love such regulation and there's no countervailing pressure for less regulation of course. For how can the companies that never come into existence impose political pressure for less regulation?