Given the way employment law in the US works, it is HR's responsibility, in general, to ensure the paperwork is in order if an employee needs to be fired.

And yes, absolutely, even large publicly-traded companies still have bosses that will fire employees because "it isn't working out" (which is the aggregate signal in the neural net with inputs such as "I don't like this person," "I think my job is harder because the company employs this person," "I believe this person's presence at the company is bad for the company," etc.). The difference between large and small companies is that large companies have more to lose if a termed employee lawyers up (and more likelihood a termed employee will lawyer up, because more lawyers will be willing to take a case that has a big-name defendant in the case name).

This is yet another scenario where the question "What are the incentives of the actors in this story" can be applied.