Is it possible to use a dead 9V battery as a 5V power supply?
A healthy battery has low ESR to support  %V drop / %rise of I = ESR [Ohms]

A dead battery has its charge capacity Q=CV with in Farads now reduced &lt; 1% to &lt; 10% of initial charge.

At the same time, dead batteries ESR rises >>10x the initial value.

A stable supply is often defined by a 10% V tolerance for some rated max current which also defines the Load Regulation error now in terms of ESR. 

So if your "circuit has minimal current load, it can work, but is it reliable, alive, suitable? 

No. It is called "dead" for a reason. But if it works for you, with no significant load except logic levels, play with it.  But this is not how Engineers specify performance. ;)it just worked)

Engineered power supplies have (at least) guaranteed specs for mAh Vmin:max  or Vnom, % tolerance error and load % V regulation error.

This has no specs. so you decide.