I've posted extensively on this elsewhere. The passage you're referencing does not use the words "the same god". You can argue that that's what it means, but that's your opinion and nothing more.
The actual language says, "[Muslims] profess to hold the faith of Abraham, and together with us they adore the one, merciful God, mankind's judge on the last day".
So, let's stipulate as to some facts:
Even if Nostra Aetate were infallible (and I don't think it is), the above is all that it could claim. Why? The Catholic magisterium's charism of infallibility extends to the teaching of Catholicism, not to the teaching of any other religion. Claiming anything like "Muslims and Catholics worship the same god" is making claims about Muslim teaching which are far beyond merely stipulating commonly known facts, and which are far beyond the magisterium's competence to make with any authority.
Which is why I don't think that Nostra Aetate 4 played any role in the DP at all.
My objection to the SSPX, aside from their disobedience, quasi-schismatic behavior (on display in this thread !!), and wrong ecclesiology is very simple:
A substantial percentage of their adherents reject the new Mass, even when celebrated according to the rubrics, as invalid. There's even, IIRC, an SSPX examination of conscience I've seen which identifies attending the new Mass as a sin to be confessed!
The proposition that the new Mass is intrinsically invalid (I'm speaking of Masses celebrated according to the rubrics, not ones full of invalidating liturgical abuses which are recognized as such by Rome) is a heresy against the dogma of the indefectibility of the Church. That SSPX & friends tap-dance around this heresy on a regular basis means that they are both quasi-schismatic and quasi-heretical.
Come back to the Catholic Church. If there isn't a parish near you that offers the EF Mass, see if you can get one started.
Folks interested in knowing what "Traditionalists" really think of the Ordinary Form of the Mass need only web search the term "novus ordo cookie" (use of quotes will help keep the search relevant). The results will be highly revealing.
So if the muslims are mistaken in that, why does Nostra Aetate continue on with the heretical statement that the muslims"together with us (Catholics) they adore..."
"Together" is the keyword that you ignore, Campion. Catholics adore God correctly; muslims don't. Catholics and muslims therefore cannot adore God together.
(Except maybe when a pope kisses a Koran or takes off his shoes in a mosque; then they really are adoring together)
“Come back to the Catholic Church. If there isn’t a parish near you that offers the EF Mass, see if you can get one started.”
Thanks for the invite, but I’ve never left it. I encourage you, however, to extend you invitation to Protestansts, Jews, and all others. Cheers!