You are arguing about semantics of residence permit vs work authorization which is not the core of the issue. If you get fired and don’t find a new employer then you leave in 3 months.

Also, it is definitely not just a formality to change employers. For example, on a blue card the new employer must prove to the ministry that they couldn’t find anyone local or EU to fill this position aka “Labour Market Test”. The position needs to be registered in a special gov database to prove that, etc, etc.

The requirements are far from uniform, because each member state sets its own policy. For example, Finland requires the labor market test from ordinary employees but not from those with a Blue Card or those applying for a specialist permit (similar to the Blue Card).