Yoshida already explained this months ago. Summoner will simply have new summons this time around.
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It's going to presumably take a reasonably active, organized and skilled group of players to conquer the primals to begin with. Chances are, if you can conquer the primal, you can conquer most of the content they'd be tremendously useful against. So it's more like a time-saver or guaranteed win in a speed run.
I imagine at some point hard mode bosses might be a step above the primals, but the reality is, you're not going to b e able to farm content if you have to rely on a 2 week cooldown (minimum) to win.
As long as no content is balanced around the presumption that someone will have a primal, it shouldn't really cause much in the way of grief. For the super elitists, people might see it as a way to trivialize content if they ever add any insanely powerful bosses a la XI's AV/PW, but if they really wanted to, they could have certain bosses that are intended to be ludicrously hard have counters for primals, much akin to how many bosses in Final Fantasy X would "desummon" Aeons, making them only an initial burst of damage, instead of allowing them to fight it out with the boss.
I'm most looking forward to busting a primal out in PvP, that would be quite hilarious, and as for anyone saying no one would play against anyone in PvP who had a primal, how would you predict that? For one, they've stated there will be no method of tracking who has the primal other than word of mouth. And for two, assuming the "battleground" style PvP is open world or queued, there's really no way you could completely avoid the chance of running into a primal.
If they're all only going to be summoned at minimum every 2 weeks, and I'm betting most people would use it for PvE (way more practical, way less fun) there's really no situation where it's going to actually cause grief in PvP. That one time some pre-made group uses it would just be a hell of a lot of fun. Heck, even if I'm on the receiving end of something like that, I'd find it entertaining. I doubt it would last long enough to completely unbalance an entire battle.
P.S. Not all drama is bad, it's as much a part of making a community strong as anything else. A little competition never hurt anyone, and in this case I think a boost like this is a healthier thing to compete over than an HNM with a unique drop.
Of course they would hopefully be able to turn the tide of a battle when used correctly, I.E. at a key moment, what I meant was that they likely wouldn't last long enough to completely roll the other team. Destroy the flag bearer with Zantetsuken before they cap the final flag, something like that.
I.E. it'd likely be an ace in the hole that could turn the tides of a single PvP battle when used properly, but nothing that would cause enough grief for a long enough period that anyone would ever feel the presence of a primal in the game was so invasive they'd avoid PvPing.