With all this said, what is a good angle/approach to implementing such a startup? What are similar examples you know of that worked in the past, or at the have a working theory behind their construction?
As an example, my town of about 400 does everything a government needs to, with a handful of people meeting once or twice a month, a website, some email addresses, a few Google docs, and a physical town hall to meet in and hold elections. Don't get me wrong, the people who run the town do put in work, but it is far more simple than the state or federal level.
Before you try to build some new solution, I'd research such simple governments and figure out what the minimum needed to operate really is, and where the limit will be as the size scales. Because while the few large government entities are the most obvious and talked about there are literally tens of thousands of small government entities to look at to fully understand the differences of scale in governments.
Look at: Small towns, Library Districts, School Districts, Sanitation Districts, even HOAs.
1. Gain the ability to veto digital projects above a certain value as being too big to be able to be realistically completed on-time and on-budget.
2. Take on some of the legal risk, enabling you to reduce the amount of red tape and allow smaller agencies with fewer lawyers to enage.
3. Build a bidding marketplace for smaller, better defined projects that are able to be tackled by smaller agencies and startups.
More info at https://www.digitalmarketplace.service.gov.uk/
A government is far bigger and complex than any big company. So there will be more slowdown compared to big companies.
One example of successful public-private partnership is India's "Digital India" initiative where the government was able to churn out platforms like Aadhaar, Unified Payment Interface and RuPay, with other platforms on the way for digital commerce for MSMEs.
Do governments share your belief that the problem you want to solve is an actual problem they believe needs solving? Democratic governments have elements of serving the people, all of the people, and limiting factors (like using certain technology) can be restrictive.