While I personally agree with your latter sentiment, the Merriam-Webster dictionary defines bureaucrat to be simply, “a member of a bureaucracy”. Where the top definition of bureaucracy is:
A government is by definition a bureaucracy, but so is any private corporation. Any organization of sufficient size becomes one out of necessity. Any time you have a bunch of people together and you need to establish any kind of pattern for how things get done, you have a bureaucracy.
What is funny is the way people use the word derisively, as if bureaucracy is inherently evil in some way. Bureaucracy is like math, it is netiehr good nor bad, it simply is. Like math, it can be used to do good or bad things.
a) : a body of nonelected government officials
or
b) : an administrative policymaking group
So really, any government is a bureaucracy.