Except what you're describing is the healer's own job description: to heal missing health and keeping people alive. If a stooge does land on The Bad™, then it's annoying but it's part of our job as a healer. It's literally the difference between us and DPS. Unfortunate effects, but that's why we're there.
We're NOT responsible for the person being hit in the first place, nor should we be made to be. You can argue that it's as much a healing tool as a shield tbh, you're preventing damage, but it's already been proven to be incredibly intrusive whereas shields are not. And worse, similar stuff exists in other peoples' kit, such as Second Wind and personal shields, but they don't deny our existence. Only cover for one singular hit. The use of rescue, on the other hand, negates there needing to be mobility tools. After all, why bother having Shukuchi or Aetherial Manip if you can just nag the healer to adjust for you?
That isn't a logic extreme you can extrapolate out of what I said, you're conflating two ideas and snowballing out of it based only on the idea that we're there to babysit.
You're only there to babysit as much as you want, because it's not up to you to rescue every single person and know exactly when it's okay or not to pull them. You are there to be a healer when required and deal damage when not. Other people should pull their weight just as much as you do. If I wanted to babysit, I'd start charging for it.
Otherwise, yeah, it's a double standard. We're not responsible for people doing well and keeping themselves alive. There's a limit to what we do when faced with what other people can do. If a tank can use cooldowns, why the hell do I have to go out of my way and strain myself for someone who doesn't want to meet me halfway? Just use cooldowns, move out of the bad. Accidents happen, and those are stuff I can understand... but they shoudl not be on me for failing to pull you away from those accidents.
Keep in mind also that you're only saying that, that it's a part of our job as healers to do so, simply because Rescue exists. Prior to SB, this mindset did not exist. It was created only bc of the skill itself, concentrating responsability on the healer and making people forget that no, it's not our job to save a guy from going to the red puddle. They should know better, that's part of the game.
Though I guess I feel this way because I'm angry at what happened to me in savage before. Being told that the whole group needed to adjust around one person that refused to learn the game and improve as to not be a detriment to the group. And would then throw a tantrum when called out on it. It isn't our job to bend over backwards for someone who doesn't want to meet us halfway. Which hurt doubly so when I did my best to overcome my own physical health to play with my groups, finding tricks and roundabout ways to handle situations, only to then see someone not do the same and have everyone adjust to her alone. It just isn't fair to 7 other people; one thing is "I struggle with this", another thing is "I won't even try".
I don't think Rescue is griefing, but it does remove agency from people and excuses something that really just shouldn't be excused. And when used incorrectly, it can be quite bitter.