While this isn't much of a company relationship thing: If you're getting paid every 2 weeks, sometimes you'll get paid the first/third week of the month and sometimes the second/fourth, which can be a pain if you're trying to keep a consistent amount in a checking account while also having bill pay and auto-transfers to savings/investment accounts.
Or, tl;dr, it makes it easier to automate money without leaving a large buffer of cash in a checking account.
I think what you want in checking -- if your circumstances allow for this kind of money-shuffling, at the least -- is more along the lines of 4 weeks of expenses, not 4 weeks of income, which would (hopefully!) be considerably higher. I'm not sure I'd call it a "large buffer," but even if you're not in the "shove absolutely everything you can into index funds and live like a pauper despite the fact that you're making $250K+ a year so you can retire at 35" camp, the more money you have in higher-return places, the better, so you probably don't want it in the checking account. (And probably don't want a lot in a low-interest savings account.)
Or, tl;dr, it makes it easier to automate money without leaving a large buffer of cash in a checking account.