Don't mind the booing too much, the majority of the time the crowds respect the Cup while it's being brought out by the handlers and then during the raise by the winning captain, it's pretty clear during the presentation the Cup itself isn't being diminished, which is personally all I care about.
Perhaps the Cup's simply been won too often on the road recently, makes the ceremony anticlimactic.
What does irk me about the presentation is how "corporatized" it feels (don't know of a better word). I preferred in the past when after the Cup was handed to the captain, the team would surround them as it's raised and skate with the holder as they parade it around the ice, handing it off between each other as they skate in celebration as a group. The way it's devolved (imo) into the current iteration, where the team stands still at center ice watching as one person takes their turn to skate away from the group, just comes off as artificial, as if somewhere along the line teams were instructed to celebrate this way so the league could have the close up shots of each player skating with the cup for advertising purposes instead of a spontaneous expression of the team's emotions.
Something about it just diminishes the team aspect of it to me, especially in some years when half the time when the camera lingers back to the group they're all half-heartedly clapping for the current carrier as they wait their turns, or aren't even paying attention as they're busy with one of their other teammates/family/interviews/etc. After the big names/feel good stories I barely feel anything when the camera guy follows the third line penalty killer as they take their short skate before u-turning back to the group, even worse when said third line player is barely cheered because they're not as heralded. Outside of when my team won because I had a vested interest, they've mostly been pretty bland as an impartial observer.
I don't even remember when it really changed, I just remember at one time that's how it was done, and preferred it. If I could change one aspect of the presentation that I'd think would improve it the most that would be it. Just my opinion of course, others I'm sure disagree.