Not internally, but at least in the US you see lots of attempts to cut budgets of government programs. Even a modest budget cut could have a big impact on the level and quality of service of a government health system.
Tons of government programs have wait lists because they don't have enough money. And other programs don't give people enough, because the program doesn't have enough to do so.
Any government run healthcare would be money constrained the same way, and they would have to do exactly the same as the insurance companies to try to save money in order to have enough to at least give the impression of being fair to everyone.
The issue here is not for-profit insurance companies, it's that there is no competition on the Doctor side of things, so costs just go up and up and up.
Those organization types are just as motivated to save money as for-profit is.