The way I look at it is this: He said they were saying it was a farming party, to me that implies experience. They know how to do the run, or want people who do, and people have win(s) under their belt. That they are good enough to farm the primal, or so they said, says a lot. As the GM's have said, it's fine to kick people for whatever reason, but they highly advise parties talk about whatever issues or strategies they want to use before starting.
As to what the OP said:
They never discussed it starting off. Never told people what role they would play, and never asked if there was anything specific that each person should be doing. Everyone learns the fight different ways, there may be groups that win with the tanks just swapping and someone else shouting out. Perhaps in another, the tank does call it out and the DPS just follow instruction without really ever learning and just waiting for the call. There is more than one way to do a fight, and if the whole team is not making sure everyone is on the same page, that's not one person's fault.
As for voice chat programs, while they are nice sometimes people can't use them. I know if my husband was asleep and I hopped in and started chatting away about some fight he'd probably ask me if I'd lost it. lol Though out of respect and to keep on the same page, the OP should have said something just to give a head's up like: Thanks for the offer but I won't be able to get on. Now that being said: OP could have said he couldn't get on or had got on and listened, but told them that he couldn't talk. Also, when you're in a party and someone says 'come on TS...whatever reason' that is usually a general to the whole party. It's not just about one person, and people shouldn't expect a special offer just for them. If they didn't specifically say XYZ get on TS, they were telling everyone they could.
There were two tanks, the party should have asked one of them to call out the orbs and they should have discussed who would be doing it. If you have two tanks, and neither are calling it out, then you can't just put all the blame on one person.
So tl;dr or whatever: It seems to me there was a general lack of experience, and lack of communication within the group on all sides of the coin. Because while OP did say XYZ did happen, and people didn't know what to do, he also never stated that he tried to communicate or help in any way and as OP admitted, all were wrong in this instance, even the OP.
But the issue is... with your OP post you seem to only be blaming them, (they can't count, they can't do their job, but it seems the OP didn't communicate in any way to improve the overall situation) but never said you're also at fault until many posts later. Over all it seems like that is the reason it failed and it's everyone's fault even the OP's.
Edit: Forgot to mention in general, I play PLD and BLM main, so I understand things as well from tank and dps point of view.