How hard would it be to found an insulin startup? This seems like the kind of problem designed to be intractable, the "busy work" of bureaucratic private health care systems.
You have to avoid the intellectual property minefield on the manufacturing side. The production of insulin and its delivery mechanisms are heavily patented and the big pharma companies will shut you down if you infringe on anything. If you want a reference for this happening you can google 'lusduna'.
Plus you have to make it through the FDA, which is its own can of worms. The 505(b)(2) is still not a cakewalk but at least it will give you 3-5 years of market exclusivity so you can recoup your costs. (Wait you can get market exclusivity for a biosimilar with an expired patent? That should sound insane to anyone not in healthcare.)
Then you have to actually manufacture and distribute your product. There are extremely high capital costs to get set up. For reference, Novo Nordisk's insulin manufacturing facility in North Carolina is 328,000 sqft and staffed by ~500 people 24/7. Just to get a comparable facility up and running you're talking about 400-500 million dollars in capital cost. Sanofi, Eli Lily, and Novo Nordisk have spent billions over the last decade upgrading and maintaining their facilities.
I would suspect it's about the hardest project you could tackle if you were a startup.
I can't speak to drug manufacturing, but I know from experience that when it comes to medical devices, you save almost 10x in up front fixed cost by outsourcing, like California is doing. You spend more per unit, but still profit, so you can build a healthy warchest while you accumulate experienced personnel to build out a more vertically integrated manufacturing process.
Medical devices are frankly a lot simpler than making new drugs. (I know they are still exceptionally challenging)
Edit: Answering myself here, they do not: https://www.vorys.com/publications-2574.html
However, many patents that are issued in association with a research grant include a clause that allows the government to use the patent without charge. I have no idea if any/many insulin patents are included in such clauses.
Now, if you closed the loop and used older insulins, you'd have good control and less issues since a computer would bear the cognitive load, but glucose control, especially in the outpatient setting, is ridiculously hard. If you think you can solve it though, give it a shot. It's a multi billion dollar market and a lot of lives you can improve.