Comes down to 3 factors.
1) Tank
2) Healer
3) DPS
1. Tank - Gear is a big factor into how much the tank can pull. I don't care what kind of healer you are, if the tank is taking around 3K+ damage a second, you aren't going to be able to keep him up very long. Also cooldown management is huge for a tank on big pulls. If he's an idiot and blows all his cooldowns at the same time, he's doing it wrong. Dodging red circles, holding threat, picking up stray adds, etc. all a part of being a good tank.
2. Healer - Not sure how well geared you are, but in order to be able to take on big pulls you will need to be able to pump out sufficient heals. Your rotation does seem a bit off, Physick with Adlo and trying to Stoneskin? This may work on smaller pulls, 3-4 enemies, but on big pulls as you suggested (going to guess around 10 or so enemies), you would be struggling greatly keeping any tank alive.
First of all you're a scholar, Stoneskin does at most 10% of the total HP of the target. Now if you have a Warrior tank who is completely geared out, he'll have 10K+ HP, which is a 1K Stoneskin from you. Since most tanks do not have this kind of HP, let's average it out to roughly 6-8K or 600-800 Stoneskins. Which does not heal at the same time, unlike Adlo which does heal at the same time of putting down a shield, and with a crit has the potential to be far greater then any Stoneskin you could cast on a tank. Pop out a Sacred Soil instead of a Stoneskin next time.
Not to say that Stoneskin shouldn't be cast before a pull, it should, but during a fight, your Adlo will prove to more beneficial.
Lustrate should hopefully be saved for when the tank no longer has any cooldowns, and is kind of struggling to survive. Basically to get him to a safe level so you can cast Adlo again. Can also be used after the tank has established initial aggro on the big pull, and you haven't cast a heal due to fear of grabbing aggro, all depends on how much damage they've taken though.
Don't forget your pet abilities. While the majority of the enemies in dungeons deal mostly physical damage, there are a few that will deal quite a big of magical damage when big pulls are involved. Fey Covenant will help mitigate some of this damage. Fey Illumination also helps out for when the tank is running out of cooldowns, and you could use just a bit of extra healing keeping them alive. Of course if you're using the other Fairy you could always use Fey Glow and just try to get your heals out faster.
3. DPS - Last but not least, there is no point in making big pulls if your party consists of a Tank, 2 Melee, and a Healer. While DRG's and MNK's can deal AE damage, it's obviously sub par compared to that of a BLM or SMN, or even a BRD. If the damage isn't good enough to kill things in a timely fashion you will find yourself running out of mana, the tank with no cooldowns, and your fairy sitting there wondering how the hell it's supposed to heal the tank through all that damage.
All in all, I feel your rotation is a bit off, but we also don't have the full story either. How well geared the party was, including yourself. The type of DPS you had. How good the tank was at managing their cooldowns, dodging red circles, holding threat, etc. A lot of factors come into play when you think of who may be right or wrong. Who knows? Maybe the tank could have done things a little differently as well, maybe only pulling 2 packs of enemies instead of 3 or 4 (or more). I don't place the blame solely on one person, there's always room for improvement for everyone.