If innovation is motivated by competition, then the government having a monopoly in governmental services means they never need to innovate, unless they receive some competition.
Because barriers to emigration are so high, countries don't have to compete with each other for quality of governance, and so it turns out that only the private sector can do it.
Your implication is that competition has to be with external agents. It’s perfectly possible to compete with the status quo.
The private sector has a profit motive at its core: make more money for shareholders. The public sector also has a profit motive: save more money and be more effective for tax-payers. They’re both examples of efficiency, two sides of the same coin.
I don't think the public sector has the motive you claim they do. I think that in practice, the actual motivation of each department of government on every scale is to draw more funding to their own budget in a massive power struggle for slices of the societal pie. There aren't shareholders in the private sense, but the profit motive is still to grow revenues/power and get them assigned to yourself.
Taxpayers don't want efficient use of funds either. From the point of view of a taxpayer, the most effective use of funds is to have them all benefit yourself. Everyone wants to maximize the piece of the pie that is assigned to them. Politicians use this as an excuse to grow the scope of government, regardless of what side you vote for. As a result, the voting apparatus does not serve to optimize government efficiency.
All I can say is that from within my department, and from a lower-than-senior-civil-servant level, the ground looks very different. It’s not for nothing that we have our design principles[1] that have ‘user needs’ and ‘do less’ as points 1 and 2. We absolutely need to have funding to do that, but specifically at GDS we’re part of the UK Cabinet Office’s Efficiency and Reform Group - a primary mission for us is to do more with less on a pan-government scale.
To argue your second point, tax payers want an easy life. If we can cut minutes or hours off your interactions with government year on year then you’ll be happier, regardless of whether your overall burden goes up or down.
Because barriers to emigration are so high, countries don't have to compete with each other for quality of governance, and so it turns out that only the private sector can do it.