I agree I wouldn't disable the ESP either until you get it fixed! I remember my old 350z was a bit twitchy when the tyres were on 3mm especially Falken FK452 - not a fan of them TBH and wouldn't use them again.
And yes, ESP brakes the car, depending on which tyre is slipping is where the brake is applied. From what I remembered, the ESP on the 350z is sensitive, I remember mine coming on sometimes when taking the long sweeping slip lane from the M6 -> M56 at 70mph, that's a bit of an experience I can tell you!
Process of elimination this one is, could be tyres, what about the alignment if you've got uneven wear?