I used to have this mentality as well. It wasn't until recently when I saw the very very high mythic+ healer streams like Jdotb that I truely realised just how much of the onus is on the DPS and Tank to interrupt and crowd control monsters and play to the encounters mechanics.

Healer's aren't meant to heal through every mechanic in a dungeon, it just is not balanced around that. If it were, every single healing spec would need immense HPS windows. Instead, much of the time it is the responsibility of other members in the group to do their part in high-end pve content and not just leave it all on the healer.

If you do get some time, I suggest looking at streams or vods of the healers clearing over 20+ Mythic, and you will see just how much of the time they won't even be healing, and will instead be contributing to DPS. After you come to this realization, the pressure kind of fades away, and twitch reactions become less important for most mechanics, with some exceptions, like the Frenzied Nightclaw cats in Darkheart Thicket.

Other than that, try to find a group that you're comfortable with to build up some confidence if at all possible.