I think you might have underestimated what Keuchel’s demands were. While probably lower now, he was reportedly looking for closer to a $22-25mill AAV. While he is better than Eovaldi, Eovaldi was the hot name at the time and getting better offers than the Lynn type deal already.
While signing Pearce and ignoring the pen seemed silly, I think Dombrowski is counting on his ability to build cheaper bullpens and has a long history of using inexperienced closers. I would not be surprised if he planned to avoid spending heavily on a closer from Day One. Whether or not you agree with this strategy, it is something he has done multiple times with multiple teams with mixed results, that obviously include some successes.
Not to mention, your thoughts at the time for Pearce was he would get multiple years at a much higher AAV. Maybe if the Sox had waited him out, he might have settled for a $2-3 mill deal. But even then, I don’t think DD was bringing in a new high salary closer. He seems to really like Brasier...