YY is most likely to take the most money offered to him. If that is from the Red Sox, he will likely sign with the Red Sox. If that is from the Yankees or Dodgers or Mets, he will likely sign with one of them. If the offers are very similar, he’ll go to the team/city he wants to play for most, and that is unlikely to be the Red Sox.
Otherwise, he is going to whomever offers the most money, just like most other free agents in the history of free agency. Is it really controversial to acknowledge the Red Sox will have to outbid/overpay to get him? We have known this from moment one. Whether he comes to visit Fenway or not, if the Sox offer him 10 and $400m, he will likely be signing with the Red Sox. If the Sox don’t blow other offers out of the water by $50-100M, that doesn’t mean they “didn’t try” or it was “all for show”, it just means they didn’t think it was prudent to do that.