To enable this, won't coinbase need its own check-collection and clearing system? Is it allowed to do this in the USA as a non-banking corporation? Also, What does "paycheck" mean here? Who gets salaries as checks in 21st century?

I think one of the important use-cases that Coinbase is trying to capture is people trying to invest in Alt-coins listed on Coinbase. A lot of people don't want to go through converting USD from banks to BTC and then pay the fee for further bridging to Alts or other chains. Once the funds are directly on coinbase, they can easily change to Alts or Bitcoin as they desire. IMO, This is just a smart additional step to make it easier to onboard customers.