I personally disagree on the Ult, but I also build a tankier Adagio than you. I tend to max it because despite the build up it has (all you have to do is build crucible) it is the one move that can effectively and decisively turn a fight completely in your teams favor. Way too often (because I build tankier) do I have the enemy targeting the laner on my team, I heal them as they run into the bush, (and since I build Frostburn they're slowed) and then they take a harsh amout of damage. Other than that, nice build!

Just used the hybrid build for my first ever match with adagio and I went 11-1-14!

For a support build on Adagio. I think Clockwork is awesome. You are right, though, the burst heal will not work until the hot is done, but that's why you're playing support - to heal others. Just keep track of who still has the hot on them and who doesn't so you can more effectively use Gift of Fire. And if you need to still support them, use that warhorn to get them out of the situation :)

I think you should rephrase "clockwork is a fools item" to "don't be a fool with clockwork" i.e. don't snipe your own heals.

In your support build you have Warhorn and Aftershock which equals 55% cooldown acceleration, but you specify that Adagio should never have over 35% cooldown acceleration. Am I missing something?

BP is broken, it truly is. I like that you have Warhorn in a lot of your builds - it is such a good item. One thing I do suggest is Crucible instead of Aegis for the Tank build. A tank should be able to help out teammates and Crucible does just that.
I like how you went through the math for the weapon build and showed your reasoning behind each build. You might want to add in what sequence to build those items.
tldr; great builds; needs more pictures.