I've pursued personally-funded research for more than a year. Some thoughts:
Think about your competitive advantages. What makes working at your place different from a university and why would a researcher choose yours? It could be they don't like to teach. It could be that they want their life taken care off and you can hire a cook/cleaner/etc. for them. It could be that they don't like writing grants. Or they have trouble winning grants. (E.g., their topic is out of favor for various reasons.) It could be the freedom to work on anything. It could be the location. It could be that they could be the most senior person at your institute. Perhaps they want to start something new. It could be that they love that you think their research is valuable.
Your own institute has weaknesses, but they can be ameliorated. Make it easy to travel to. Putting it near a major university may help earn academic traffic and provide for collaboration.
One way to structure the institute is to have one good mind and many followers on. That is easy to bootstrap, once you get the good mind. The other sustainable structure is to pick a topic and have the largest most-focused group working on that topic. People will come for that, but getting it off the ground will be difficult. Perhaps you can a few good minds right after they get their PhDs and build around that team.
If the key is the people, I suggest you talk to the people. Find out what top researchers don't like about university life.
If you need a topic to focus on, I did research on formal proofs --- writing a mathematical proof on a computer. Every other industry has moved to digital, computer-aided design, except the field most suited to it! The way that mathematicians advance their careers in academia is working against the adoption of the technology. A place outside academia might make more progress in the area.
Think about your competitive advantages. What makes working at your place different from a university and why would a researcher choose yours? It could be they don't like to teach. It could be that they want their life taken care off and you can hire a cook/cleaner/etc. for them. It could be that they don't like writing grants. Or they have trouble winning grants. (E.g., their topic is out of favor for various reasons.) It could be the freedom to work on anything. It could be the location. It could be that they could be the most senior person at your institute. Perhaps they want to start something new. It could be that they love that you think their research is valuable.
Your own institute has weaknesses, but they can be ameliorated. Make it easy to travel to. Putting it near a major university may help earn academic traffic and provide for collaboration.
One way to structure the institute is to have one good mind and many followers on. That is easy to bootstrap, once you get the good mind. The other sustainable structure is to pick a topic and have the largest most-focused group working on that topic. People will come for that, but getting it off the ground will be difficult. Perhaps you can a few good minds right after they get their PhDs and build around that team.
If the key is the people, I suggest you talk to the people. Find out what top researchers don't like about university life.
If you need a topic to focus on, I did research on formal proofs --- writing a mathematical proof on a computer. Every other industry has moved to digital, computer-aided design, except the field most suited to it! The way that mathematicians advance their careers in academia is working against the adoption of the technology. A place outside academia might make more progress in the area.