Ok, I can see where the confusion comes from.
Strictly speaking, the FSSP doesn't meet the definition of Trad, since they insert the novel idea (and go against the original Trad thinking that we have the right to attend the Tridentine Mass, permission or no permission, because Quo Primum gives us permission).
Long story short, Trads have (either literally or figuratively) a copy of Quo Primum hanging on their living room wall. It is the foundation of the whole movement. Further, we believe that no one can force us to put our Faith and our souls at risk -- that we have to save our souls and continue living our Catholic lives (which includes Mass and the Sacraments) -- Crisis or no Crisis.
The FSSP introduces the novelty that we need, or would PREFER TO HAVE, permission from the Conciliar Roman authorities. Most Trads dating back to the beginning of the movement would disagree. If they go against one of the core tenets of the Trad movement, how can we call them Trad? At least strictly speaking.
The status of their Latin Masses is obviously different than the Novus Ordo. As long as you have "validly ordained" covered, and they are saying the Tridentine Mass, then you have a valid Mass. So the status of FSSP is somewhere in between a Novus Ordo and a "fully Trad" Mass -- though it's closer to the latter.
The million dollar question is: why stick with an official Trad (tm) chapel in the first place?
Well, I would answer that there is a certain desirability and comfort to have that stability, to have the same, stable position as those original Catholics who first left the Novus Ordo and left to find Tridentine lifeboats in which to ride out the Crisis. I'm sure this point could be elaborated upon: it's more "traditional" like the Faith itself, it shows that we haven't "reacted" to anything since then (for example, a carrot held out by the Conciliar Authorities), or changed/given up any principles of the Faith, etc.
If it wasn't needed by the original Trads, it isn't needed by me. Because the Faith had to exist at all times. If we really needed permission to be faithful to the Catholic Faith pre-Vatican II, then the Faith wasn't practiced perfectly anywhere from 1970 - 1983 (or Ecclesia Dei in 1988, or Summorum Pontificuм in 2006). I'm not willing to accept that.
I think it's safer, from a "sound lifeboat" or a prudence perspective, to recognize in myself 100% of what was found in the original Trad movement, and nothing more.