You're vastly overselling Rescue.
I love rescue, don't get me wrong, but it has a lot of things holding it back that point towards its clunkiness, and it's not fair not to mention them when highlighting its uses.
Its slow activation time and slip-n-slide nature of the pull greatly reduces its effective window. A lot of the time you need to be nigh instant in deciding to use it, otherwise a person will still be caught in the AoE. Royal Menagerie comes to mind - and if someone is already sprinting towards you, a rescue doesn't actually save them much time. If done at max range, perhaps a second. If less than that, even less. It's still enough to get some saves in, but this aspect alone greatly hinders it.
This is on top of it not pulling someone *directly* on top of you, but in front. You've got to get that much further out of an AoE to pull someone out of it. Then, its range itself is problematic in bigger arenas like 24 mans have. Quite often in Ivalice the person I want to rescue is already running the opposite direction, and once I'm out of the AoE myself it's already too late.
Then we get into the Role Action nature of it. Simply speaking, it is a gamble to take it over other abilities in certain content. The default combo of Swiftcast, Lucid Dreaming, Largesse, Protect, and Esuna is modifiable based on encounter, yes. In 8 mans if Esuna is still needed, only one has to take it, and someone can drop protect too if they feel like it. However, the paltry DPS gain of cleric stance and the recently buffed surecast are extremely viable options in these scenarios. While people can drop Largesse, too, that's a guaranteed healing boost to use at your own discretion. Eye for an Eye is reliable mitigation in dungeon pulls after you drop protect, too.
Rescue is an option as well, sure, and I'd love to take it more. In situations with friends, I don't slot it often because that 10% chance they're going to screw up a mechanic I can rescue them from in an attempt is just a gamble without payoff. It's not that "They screwed up so they should die" - it's that I can't justify the opportunity cost of it. I'm going to do everything in my power to keep my raid alive. Even in savage, the unfortunate reality is I'd typically find cleric stance more useful for us and the clears we had during the boss's enrage tend to agree with me more than the times I could save someone with a rescue. I get that you're on the safety side of things. I tend to be, too - my DPS is never going to be as high as some of the healers here. My piety will likely always be higher as well. I'm all about survival as much as possible so people can get accustomed to mechanics quicker. I would rather save LB3 on Shinryu EX for the enrage rather than use it to push him into it earlier specifically to ensure the clear over maybe a quicker kill. Our reality is that DPS matters that much in savage, though. I simply cannot choose to ignore it in favor of something I might use every 8 pulls of a boss until enrage is more of an afterthought.
There are fights I -can- justify rescue in and those are often in duty finder content such as normal trials or alliance raids, where a lack of Esuna won't often kill (minus doom) or lead to an enrage wipe, or a lack of protect after a raise won't severely hamper someone's survival rates down the line. Make no mistake I love pulling off a well timed rescue, and it makes me oh so happy to pull someone with lightning on them in ramuh over to the stunned person praying someone stacks on them so they don't get to idle the next 30 seconds. I will gleefully pull the DPS stacking with the tank on Ravana who is eating laser cones to the face for 15 seconds. I've used it on Tidal Waves in Shinryu sometimes, as well as plenty on the divebomb. It has the added benefit of helping teach people that maybe they should be over here. I've had it available to rescue people who are running giant orbs away on Diabalos or lasers on Cloud of Darkness should they be a little slow, but it hasn't come up yet. Same idea for being able to pull people into our group's little "kill an add" zone on fights that split you up, but again, hasn't come up yet...very surprising on that one.
I -like- Rescue, not just for DPS. It just leaves a lot to be desired at the moment. Making it quicker, perhaps extending its range, I'd say lowering the CD too so it can be up for most major mechanics rather than every other one - they'd all go a lot towards making rescue a better choice in my eyes. Even then, it still likely wouldn't be taken on the raiding scene unless people can reliably get a decent boost of out it like Sebazy mentioned with O4S due to the nature of high DPS requirements in savage. It's simply our reality.
Now, if they let me rescue stunned or rooted players? That'd be pretty great, and situationally useful especially for things you said you enjoyed doing, like undermanning content. You can pull someone out of LB animation lock after it completes, which I've rarely had the chance to do in a meaningful way but it has come up. Would be nice to use on stuns or petrifies. Heck, resucing someone stunned on the tail in Shinryu Ex would feel nice....if not risky as you don't want to pull someone off the edge, but still. I'd also appreciate if rescue didn't go on cooldown if it misses. Ever tried to rescue a holmgang warrior or a tempered will paladin? Alas.
I just don't think it's fair to lump everyone who doesn't take rescue into the category of DPS-obsessed-speed-kill-gogogogo'ers. I mean, Sebazy in particular is one of the ones here who was raiding in the days where it was 100% about survival and enrage mechanics simply didn't exist, being able to kill bosses with 10 people that usually took 30-60 if you all stayed awake and focused on living more than anything else.