This is what happens when you treat humans as resources.
Obviously no middle manager is feeling that specific stress at Google, but the higher up you go the more your issues and stresses involve keeping the company afloat.
The guy was ill, he should have been put on sick leave not a "performance improvement plan".
Sometimes you know it must be done but you know the person is nice enough but just not competent, or can't do the job, or the job has changed and they're getting fired for no fault of their own.
There are those times when you fire someone but you feel guilty that, as their manager, maybe you could have done better with them but you didn't have time, were too distracted, too tired.
In companies that get big enough, the percentage of people getting fired for cause gets higher. Then firing people becomes part of a routine. If you let your feelings get involved, it will drive you crazy, so you try not to let that happen.
Regarding the guilt, I think you're right -- a lot of people don't realize that every day spent trying to work with an unproductive employee/coworker is a day lost by the productive one. At a certain point or company-size you can't continue doing it.