Ultimately, Davis chose Klutch because LeBron and Paul won the recruiting war. They tried to force a deal earlier and it didn't work, but ultimately they got their man because no team was willing to bet the farm on a potential rental and call their bluff.
All the players know each other. If Kyrie truly was committed to Boston and wanted to team up with Davis IN Boston, he could have easily recruited him. For all we know, he probably did but either lost that battle with LeBron or wasn't willing to recruit because he wasn't sure/already decided secretly in his mind on whether he was staying here.
Knowing this, Ainge could not pull the trigger on a deal, though he might have still done it if the asking price wasn't so extreme. The Lakers 1) got lucky with the draft lottery, 2) had intense pressure to get a deal done considering the bad press the organization has endured the last few months, and 3) had even more confidence due to the Warriors' injuries leaving the West wide open next year. It's not totally fair to expect the Celtics to have predicted any of those factors, though due to the Kyrie technicality they couldn't deal for Davis during the season anyway.
The final haul the Pelicans got was humongous and it's very possible the Celts dodged a bullet not trying to match it with something resembling a competitive offer.