However, for the problem as stated -- indexing, training -- I think private accounts are fine.
I'm a "but..." because private accounts are kind of frustrating on Mastodon. When you reply to a followers-only post on Mastodon, for example, the response is legible only to *your* followers, not the followers of the person you replied to. So, the conversation diminishes rapidly. This is a bug.
If you have a fediverse account, you can quote this note from your own instance. Search https://cosocial.ca/users/evan/statuses/115663380268744285 on your instance and quote it. (Note that quoting is not supported in Mastodon.)