They didn't have to pay it directly, but it is common for companies to have contingencies to pay employees 60% or more of their base salary "to do nothing: Amazon was already paying for, in part, long-term disability insurance, but the employee, manager, or - most surprisingly - HR were not in the headspace to even think of this, I assume because the default mode of is adversarial.
It’s the employee’s responsibility to engage STD and LTD. Everything in this article the author was complaining about was either his responsibility or due to poor choices he made.