I wonder if it is possible to only have it count the unread posts of someone who has already visited the thread at least once.
Of course you would only count for users that have already visited the thread once. Only for users that have visited the thread before an entry exists in the threadread table.
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That way, it partially acts as a thread subscription.
Partially.
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Edit: I wonder if there is any similar functionality built into the thread subscription system?
No. The thread subscription system uses the same data as the unread posts system.