It's also important to remember that most lobbying is not taking a Senator to lunch, it's emailing/calling their staffs with information about a particular piece of legislation. Staffs of these politicians deal much more with lobbyists day to day than their bosses.
>we couldn't bring doughnuts to a morning meeting with NASA because they're government officials and that could amount to bribery.
Were you a government employee? If so were you in a different agency? Can NASA employees bring doughnuts for other NASA employees?
Senators are people though and they have to eat, so maybe having a lunch meeting with the senator would be okay? Paying for a reasonably priced lunch for the senator? Say $50? Each? How about a $300 lunch? At the country club? with a round of golf after? Dinner? After dinner entertainment? And what if that entertainment "happens" to be strippers and cocaine?
The senator needs to get re-elected though. So in addition to political donations to their campaign directly, donations could be made to "totally definitely not coordinated" SuperPACs supporting them.
That's before getting into the fact that lobbyists have no obligation to present a fair and balanced picture of the situation. It's easy to sell a narrative that doesn't reflect reality if there's no one to challenge you; you can just leave out any and all inconvenient facts. Additionally, huge corporations are able to threaten layoffs of the senator's constituents.
To be clear, some of this stuff is highly regulated. Eg we couldn't bring doughnuts to a morning meeting with NASA because they're government officials and that could amount to bribery. Lobbying just has a very complex set of laws to skirt - and if you're a professional lobbyist, figuring out how to skirt those laws in order to buy influence is literally your job.