But if they would cut most of those quests and added a proper main story with cinematics for each world, it would make a huge difference.

I read this a lot and its a sad misconception that prevails on these forums.
First of all the equivalent of the "cutscene-heavy" side-quests are the zone quests you have per zone. You know..the one that starts with the war table "discussion" and the actual cutscene with Harding talking to you etc...
Those are the zone quests that accompany the core quest. They serve to animate the game world, much like the cutscene'd quests (Valena, Danyla, Burkel etc..) in the much smaller Origins.
The fetch quests (which are in plentiful amounts) are more equivalent to the relatively basic Chanters Board/Irregulars Post/Collective Satchel quests.
Their purpose is also slightly different, as the quests in DAO were more like a checklist to complete throughout the game. The ones in DAI are more zone-centered and serve as exploration incentives. They exist in bigger quantities because the world is so vast. One is not expected to complete them all in one playthrough unless one is so inclined, much like the DAO ones.
Second, you cannot just "add more to the main story". There's a cascade effect with the work that needs to be done to actually implement the writing, not to mention the potential re-writing that may occur halfway into development. Things even have to get cut!
If they could write more they would have, but you have story branching/continuity, asset design, quest-zone design, dialogue recording, animations, cutscenes, etc... that follows the writing which is a crucial first step in RPG story-based game design.
You can't even create the zones without writing. How would you design a village if you didn't know it was going to be attacked?
Then there's the fact that resources spent on zone populating/sidequests are incomparable/insignificant compared to those spent on actual heavy-duty content.
The side-quests you see are essentially created after all the main story / companion / zone quests are done.
They are pretty simple to do really.
Kill Quest:
1) Create an NPC
2) Write/Record one line
3) Add item to a mook inventory
Voila. No cutscenes. No dialogue. Little testing required.
If you hypothetically do not implement any the side quests in the game and try to make something out of the spare resources, you'd gain very little in exchange for sacrificing the entire premise of an exploration-based game. It's a little short-sighted to ask BioWare to cut it all out when many players enjoy it. What makes others people's fun less important than yours?
Prudence would tell you to just do as much as you need to per playthrough (unless you are honestly having fun) and just check out what's left on consequent playthroughs, but you can also force-feed yourself all the extra content and get burned out too.
Frankly, you are only sabotaging your own game experience by trying to complete all the sidequests per zone in one playthrough.
I know I'm not. I'm not a fan of doing all these side quests and I skipped A LOT just exploring.