Which meant that we kicked him 3 times in the same ffxiv dungeon
I'm not entirely sure if we're getting the whole story from the OP if this has happened multiple times, though I'm sure the party that he went into detail about was probably a misuse of the vote kick system. I will concede that there is a definite problem with the vote kick system and people who abuse it with little to no repercussions, but I also believe that these incidents are pretty few and far between. On a side note, keep in mind also that it only takes 2 to kick someone in a 4 person party, which means it's entirely possible that one of the three are innocent.
I personally have been on both sides of the problem throughout the years and two major incidents come to mind. My first experience as the kicker was with a gladiator tank in DoM gear in Dzemael Darkhold who wouldn't use Flash and then cursed out anyone who tried to give him advice stating "you focus on your job, and don't tell me how to do mine." This was back when Tanks could auto-queue and get back into the same dungeon they were kicked from, which meant that we kicked him 3 times in the same dungeon. When he got back, he would do nothing but stand around, specifically trying to start arguments in the party log. This person deserved to be kicked because of the fact that he was specifically bad mouthing people who were trying to help him and refusing to participate even when he returned. Of course, even if he peacefully accepted our advice, he still would have needed to leave the group in order for us to complete the dungeon successfully since he would not have been able to survive much less hold threat after the first boss fight.
The second experience in question consisted of an incident back when I tanked the Lost City of Amdapor (this was back when this was very new content). I was tanking with the absolute minimum base line gear, and since I was on the PS3 it took forever for the digital keyboard to pop up when you wanted to type. The White Mage and Summoner in the party said not to open any chests because they thought that any of chests could turn into a Mimic. Unfortunately I was mid-pull for this and didn't want to lose aggro since my item level was so low. This unfortunately became a problem because the Summoner was constantly pulling more, and keeping us in combat making it impossible for me to both play and get a message in word wise. So instead, I thought that I could show them that I know which chests are mimics and that might buy me some time so I can message them that only specific chests are mimics. Unfortunately, immediately after I opened the non-mimic chest they vote kicked the other dps in the party, thinking that he was the one who opened the chest. I stop in front of the Gourmand's gate and tell them that not only did I open the chest, but that not every chest can be a mimic, which led to a small argument. Unfortunately I was only able to get 2 messages in because the Summoner who was pulling during the entire Duty decided to pull the boss. Keep in mind that none of us are geared and the White Mage is not going into cleric stance to do damage, so when they do this, I know that we cannot beat the Gourmand since if he swallows the Summoner, I alone cannot break the mouth which means that it's only a matter of time from there. I personally stood outside the gate since I told them we need to wait for a 4th, and they insistently remained on the boss. Unfortunately for them, the scenario played out exactly as I thought, and the White Mage didn't know he needed to break the mouth, instead trying to heal the Summoner within the mouth of the Gourmand. Of course, I was kicked, though that's not to say I wouldn't have still tanked once we got our fourth, they just bullied the fact that they had 2 votes towards kicking me and tried to impose their way of doing things with it (which is Vote kick abuse in a nutshell).
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Ahead of the curve: I don't know if this is the wrong forum for a post like this one. I haven't found evidence for or against it.