I've worked both and, anecdotally, I didn't notice a real difference in efficiency. Public sector was plodding and methodical; private sector was faster but thrashed around wasting energy on the latest Top Priority project. Lots more tech debt in the private sector too. Speed isn't always efficient.
Methodical to their own ends without having to produce anything that even one member of the public wants, compared to thrashing around wasting energy on the latest Top Priority project that still has to actually make someone want to buy it. There is no comparison. Being methodical is not a value in and of itself, a serial killer can be methodical, that does not make them better than a plumber who is not particularly methodical.
The public sector is often inefficient and insulated from market forces but that doesn't imply that what they produce doesn't meet the needs or desires of citizens. And I've done a lot of work in the private sector that would never in a million years have an impact on the bottom line.
Inefficiencies are everywhere, denying the simplistic stereotype that that the public sector is unusually inefficient.