Not answering your question, I would expect the main contributing factors to be the same as everywhere.

One man's innovation is another man's loss. This is why power distribution affects conditions for innovation - people with power always fight against innovation bringing them loss.

A libertarian society is better than a corporate society then, and a corporate society is better than an authoritarian society.

Then there's the incentive for innovation - if it brings one power, then one will work for it, and if it doesn't - less likely.

This is why a libertarian society is worse than a libertarian society minus some patent protection, but better than one where patents are strong and do not reflect inventiveness and are used to gatekeep markets.

This is also why China is more innovative than Russia - in China some efficiency in actually making things makes one more powerful, but in Russia power is purely a matter of capturing it.

Political parties calling for deregulation usually in fact call for token deregulation in some areas and more regulation where their corporate sponsors need it.

Deregulation in patent and IP law is a good thing. The thing is - it's not the same as most other laws, it's the fight over definition of property on an enormous amount of value. It was treated without sufficient attention, so now it's pretty bad.

I think any real change in that would require something similar to a revolution. Everywhere, especially in countries home to corporations built on such legal framework.

startup capitol of the world

Sheer number of startups is I think a bad metric (by the way, answer to you question may actually be China and not USA: [https://hbr.org/2021/05/chinas-new-innovation-advantage](China’s New Innovation Advantage)). That is because my impression of startups is that most are reheating an existing idea in a different container. So it becomes a game of rebranding and finding a different enough costume so that the idea sells. Take for instance startups on AI. And this is one of the instances how capitalism actually hurts innovation. People create these startups not because they like innovation or they are curious about developing something. They just try to have their lucky break (American dream!) by using an existing idea. This creates a sea of copies of an existing idea, inflating what you might actually perceive as the amount of innovation in today's world.

How does one set of national policies stack up against another when it comes to the number of patents or successful businesses per Capita

You are looking at this question from the view of tech and empirical science. I agree, tech and empirical science wise innovations generally or almost exclusively take place in wealthy and powerful countries. And I think in today's world top ten most powerful countries that you can think of will likely be benefiting from capitalism in one way or another. For instance China is currently the leader in about %80 of the tech that is rated critical ( [https://www.theguardian.com/world/2023/mar/02/china-leading-us-in-technology-race-in-all-but-a-few-fields-thinktank-finds](China leading US in technology race in all but a few fields, thinktank finds)). I would not however claim that China is a purely non-capitalist country. Perhaps quite on the contrary, it is one of the countries that has benefited most from capitalism in the last couple decades or so. Therefore this does not really refute your argument. However this points at a confounder: power and wealth. To be able to answer your question conditioning on this variable, we really need a country that we can consider genuinely non-capitalist but still has a lot of wealth and power or at least some breakthroughs in a scientific field (Do you know of any? I don't, maybe medical advancement in Cuba?). It is quite likely that capitalism brings wealth and power to a country which then causes scientific and tech advancement. Therefore they maybe correlated through this confounder, but the latter might exist without the former. And indeed history of human kind is full of mind blowing innovations that existed outside the frame of capitalism in much older times, but almost always in countries that are the most powerful/wealthy of the time. From the historical perspective, power and wealth might also actually be a confounder: tech and science advancements of powerful countries are more likely to endure then those which are not, creating the Europe centric view of science history! Indeed if we look at other fields like arts, sports, theoretical sciences, maths in recent times, there are many examples of non-powerful/non-wealthy countries producing amazing pieces of work or people who have really rocked the foundations of some of these fields.

"Capitalism breeds innovation"

Finally I would like to point out one detail about this quote. The general idea behind this point of view is that capitalism creates competition which drives innovation. This is the bullshit I am calling on this sentence, not really that there is some correlation (but not necessarily causation) between capitalism and innovation. Capitalism does create competition because as pointed above it motivates people by the possibility of becoming extremely rich. And therefore people spend more effort and energy on creating something that will make them rich. This however does not mean that they necessarily try to create an original idea in order to achieve this. Many times it pushes people to take the shortest path to achieving their dream which is taking an existing concept, modifying it slightly and concentrating most of their energy on rebranding, pr and advertising so that it takes a hold. Real innovation can be painstakingly difficult and extremely time consuming, which is a deal breaker for most of these people who want to get rich fast. So for every genuine innovation we have, hundreds of copies of it suddenly pop up in the world which might create the illusion of more innovation but it really is people who are trying to get rich, i.e people who are in the innovation field for all the wrong reasons and therefore are not really likely to create anything of substance.

Big tech and science innovation wills always require some accumulation of wealth and power. However if people actually lived in a world where they had easy means to a secure and comfortable life and time to actually spend on their hobbies rather than earning a life or dreaming of becoming rich, I strongly believe that the amount of smaller scale innovations we get would still exist comfortably without a capitalist framework (sorry for the MUN style ending).

Meh, startups and businesses are capitalist organizations, and I think the idea of patents is questionable outside capitalism, so these wouldn't really be a good metrics. I'd guess the richest countries "innovate" the most because they can support more risky endeavors. The U.S. is the capitalist imperial core, so it probably innovates the most. Other capitalist nations like Haiti, probably not so much.

The best measure of innovation would probably be something like scientific publications. China wins by raw numbers, Vatican City wins per-capita (???).