At private companies the bureaucracy is constrained by market pressures

I disagree- you'd be amazed how inefficient you can be and still be profitable. Lots of very large companies are being strangled by their bureaucracy even while remaining at least somewhat profitable (generally the existence of a huge company is all-in-itself a barrier to entry for competitors). I've worked for a surprising number of companies that have the basic problem of "I used to be very profitable, but now I find I'm slightly less profitable despite selling more products at higher margins." Even worse, I've seen attempts to solve the problem derailed by the same management apparatus.

A former boss was fond of blaming MBAs. He had a saying something along the lines of- the core problem with MBAs is the idea that you can good at "business" without being good at any particular business. MBAs march in, say "we need to quantify these decisions" and add a ton of process (which invites the managers in). A decade later, they notice that despite generally better conditions they aren't as profitable, they higher some big data consultants to come in and we say things like "you are spending $x+100 dollars to better quantify decisions that are only worth $x, and thats not even counting all the time you waste for all the paperwork that the process requires."