It's great that you seek others out to help you contribute more, there's always room for improvement and your friend is right. We all start somewhere.
And I second post its near your monitor to keep track of important things. I use them all the time.

So I'll adress a couple of things in order of priority:

1) Uptime
It's okay to press the wrong button, it's better than not pressing buttons at all. That is always your top priority when trying to improve: aim to keep your GCD rolling at all times. As a SCH you have Ruin II so if mechanics tend to distract you, it's fine to switch to non-stop Ruin II until the mechanic is over (e.g. for positioning during arrows, numbers, dodging Cataract etc). The biggest dps loss is not using a GCD at all.
You have often gaps between your casts even outside of mechanics, the question is why?
My guess is that you overthink it by wanting to nail everything at once and ending up neglecting the most important thing.
As Sebazy said: don't overthink it, press buttons. The details will come with time as long as you're aware of them but for now, just press buttons.
The goal is to slidecast as much movement as possible but as an intermediate step, abusing Ruin II is fine.

2) Weaving
You often clip after Broilt. Remember that after every Broil you can only use one oGCD. Not more. Every time you clip, you lose uptime and 3 clips are roughly 1 full GCD lost.
If you have trouble with this because you panic and want to press all the heals at once the moment HP bars move to the left, switch to Ruin II until it's over. It will become easier with time to properly stagger your oGCDs and neatly weave them one after another after a Broil but for now I'd advise abusing Ruin II but be aware that it's just an intermediate step and don't stick to it just because it's comfortable.
Your final goal should be to reduce Ruin II as much as possible by weaving after Broil and slidecasting.

3) GCD heals
You currently have the habit of using Succor during pretty much every aoe cast. Keep in mind that the only purpose of GCD shields is to ensure the party's survival if everything else has already been applied but would not be enough.
The party only needs to survive the hit and every Succor that wasn't necessary for surviving the mechanic or consecutive hits is essentially like a dropped GCD even if it counts for your uptime. That is a bad habit I'd try to get rid off and there aren't any good intermediate steps like for weaving. It's a binary you either used it or not and has a lot to do with how comfortable you are on healer and how comfortable you are with seeing low HP bars.

There are more things but I'd focus on these 3 in the order listed for now.
I know that constantly relying on GCD heals as a comfort crutch is a difficult habit to get rid off because it eases the mind, it makes you feel safer and like everything is under control when HP bars are full and with a yellow bar on top of that. But if it wasn't necessary for surviving, it's still an empty GCD. However it'll probably get easier to drop GCD heals once you are more comfortable with uptime in general and more confident with weaving heals properly and in time, which is why I'd put it at lower priority for now. And PF is not the best environment to get rid off this habit either.

Once you feel like you improved with this, you can go into more detailed improvements.