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As long as you transfer the website correctly, the IP address's geo-location remains the same, and you use a reputable host, I would say you are fine. Site speed would be something to consider when shopping for a new host because search engines use this in their algorithm, and fast sites create a great user experience, which is good for everything including SEO. Also, changing a website from public to private has no affect on rankings.

If you really would want to do something about it, I would say that you will be better off returning 404's, as it is less likely for those bad links to carry any (bad) weight if they point to non-existing pages then it is to establish a relationship between websites by 301'ing. This assumes that you will 301 the old pages to a website not owned by you, as I don't think you would want bad links to one of your other websites. And because if you 301 the old pages to a domain not owned by you, in effect you are establishing a relationship with a total stranger who may do any number of bad things on its own, which in effect could harm you later. But if it was up to me, I wouldn't do anything about those bad links. Just try to get new links of better quality and they will very soon outweigh the bad ones. Also, it is quite natural for any kind of website to have low quality links, so IMHO there really is no need to do anything about those old ones, just build better new ones.