The timeline suggests they went to Mookie first with the $300 million offer. When the negotiations failed, they went to Sale and Bogaerts. Sale and Bogaerts didn't sign their extensions until right before the 2019 season.
That's why it doesn't really make sense to say Sale's or Bogey's money prevented them from keeping Betts. If they had worked out a deal with Mookie, they wouldn't have kept Sale or Bogey.
They told everyone this after the 2018 season, in fact. They said they'd love to be able to keep those three but probably wouldn't be keep all of them.