I'll keep the advice specifically tailored to things that'll help time those keys, since it'll let you focus down on some more niche things. Your play as a whole is relatively fine, barring random issues such as getting clapped in that first pull of Workshop (I wager it's from either getting hit in the back or, what seems to be more likely, melees getting synced from an AOE stop and then them all slapping you at once)
Specifically, the main thing likely costing you timing keys are the large numbers of deaths in each of these keys. There's some that just flat-out you can do anything about without just actually playing multiple accounts, like the Pala who just stood in the first lieutenant's AOE circle in Priory and died, but there are a non-zero amount of deaths from interruptible casts, like void bolts in Rookery. While you're generally carrying the interrupt counts in each of these dungeons, you can actually eke out even more. When shield charge is coming off CD, you can actually leverage it as another stop option as well as damage/SB uptime/etc. So you can actually sometimes help catch casts that would kill party mates by being willing to adjust SC targets on the fly like that. I actually personally have SC on a mouseover macro so I can just hover over a casting nameplate and catch them.
You'll also want to have a bit more awareness of controlling packs as a whole; you'll want to learn to be wary of things like melees being synced and how to deal with that (generally stopping a single mob or stutter stepping so they get into melee range at different times from you) or knowing which casts come at the same time or can't (For example, Paladins in Priory will always chain Sacred Tolls back to back, so the more you have, the more bursts of AOE damage the party takes, or whether bolt spamming mobs can target the same person at the same time). The things that will absolutely kill people are large bursts of damage they can't do anything about number wise, so you'll want to avoid the situations as much as possible. Like I mentioned above, enemies all having their melees synced will generally murder you, so don't let that happen or you die and it's likely a wipe, or hold interrupts to prevent multiple casts from hitting the same person at the same time.
And finally, the advice that you just really don't want to hear
find better groups/pugs/network to find people to play with.
The DPS you're playing with from what I can see aren't pulling their weight damage wise half the time, which costs you a good chunk of time through a run. The deaths are really the biggest issue, but anything you can do helps.