Selene is not automatically better than Eos. That is outdated information first of all.
The monk didn't have to have a healer at 60 to know that pull was healable in your gear. The notion of that is ridiculous. Anyone who has run a decent amount of neverreap should know healers are capable of healing multi pulls in 170+ gear. Getting more gear will help yes. But I was multi-pulling with my FC healer at HW release.
Yes you absolutely should be able to heal a tank through a multi-pull. It is a skill all healers should have and you will need to put your DPS on the shelf temporarily/dial it back to make sure you keep the tank up.
If you cannot heal a multi pull in neverreap there is no way you will be able to heal the extreme primal fights or savage. Even A4 normal requires intense healing by comparison. Playing a class at sixty is about bettering yourself. So even if the situation wasn't savable with your current skill you need to ask yourself what you could have done better instead of trying to shift blame. Regardless of who fucked up in a given situation you should be taking some personal experience/betterment from it.
Honestly I don't believe you about the conversation on the cooldowns. Any tank is using some form of defensive cooldowns on a multi pull so I find it hard to believe that actually happened. Maybe the fight happened fast enough he didn't pop any. Regardless it sounds like you were way slow on the uptake and not ready for the pull. A pre-pull adlo could have easily given you time to dot>bane> and swap back to healing quickly.
Don't get discouraged too heavily. Just get back in the saddle and try again.