I think that still runs into the problem of the first state being like really conservative. I think if ur looking to like change the system as little as possible and improve the primary a bit maybe you could find like a low population state that also leans heavily in your favor typically in the general and start there? That way its a small test run election with your actual base. Or maybe do like city primaries? Like NYC, LA, etc have their own primary elections, party sponsored events where all the candidates get to speak, and then broadcast them nationally. Then do a big national primary a few weeks or months later?

But tbh i think the main issue is that the primaries atleast for the dems are generally just rigged anyway. Youve got like super delegates, and back door deals to stop any left leaning candidates.