Based on what I’ve seen and heard (luckily, not experienced), I think it depends on the company. Smaller ones with older management may be both more reliant on the labor and more used to lifetime employment. I suspect there’s probably some maximal fuckiness point. Most companies aren’t like that, but resignation in Japan has a lot of stigma even if everything goes well, so many people will use these services just so they don’t have to deal with it.
The way the culture works there’s no way for the managers to be anything but unfailingly polite to an external party that calls to resign on behalf of the employee.