Yes. You're getting the Brits (covenants) mixed up. The other source of your confusion is a re-jiggered definition of salvation.
By "saved" let's translate that into Jewish worldview and say salvation is wholeness: physical, emotional, mental and spiritual.(not just spiritual - and not just 'grace'). In Judaism wholeness occurs once we choose good over forces that distort the Oneness of G-d and people. It is partly merited (we take the action) and partly bestowed (the benefit is much more than our actions).
Now for the Real Deal:
Rav Moshe Chaim Luzzatto (ref below) says we do a little bit of good, and G-d does a lot. We must merit salvation or wholeness, and then G-d's mercy kicks in to complete whatever was missing. We must show that we are willing, and G-d will provide the power to finish. (contrary to the Christians contention on this answer that one must keep 'all' the Law. We are never perfect until G-d makes us perfect)
Yeshayahu (Isaiah) 56:1-8 in context, starts back in Yeshayahu 54 & refers to the time after Mashiach come - neither the Noahide Covenant nor the Torah Covenant. All of that is describing a New Earth, which is a time and covenant of Holiness to come of the highest sort.
Torah notes a number of covenants "Britot" with G-d which lead to G-d and wholeness or salvation:
After Adam lost their initial position of Keepers of the Garden, and co-creators of the Earth, the first Brit or Covenant G-d makes with humanity is through:
1. Noah to never destroy the earth again by water. The sign is the rainbow, and the commandments are the 7 you cite. Seven as the number of evil kings replaced by 7 good kings in Canaan. This covenant is for all the inhabitants of the Earth, including the Earth itself and Jews. Jews are bound by these as well as Righteous Gentiles. Follow these and you will be 'saved' or made whole and have a place in Olam Haba, the World to Come.
2. The second is "Brit Milah," or Circumcision, that G-d made with Abraham. You don't have to do this one as a Righteous Gentile, but it's not a problem if you do. It is a covenant to wrestle with our evil inclination, and many non-Jews do this as well. Follow these and you will be 'saved' and have a place in Olam Haba.
3. The next "Brit" is the "Brit HaTorah," the Covenant of the Torah negotiated by the Jewish People with G-d, saying "We will do what you command", even without knowing what we were getting into (faith). You don't have to obey these 613 which include Shabbat either. We do since we said we would.
During the exile, all the major prophets - Jeremiah, Ezekiel, Isiah and others promise we will not be forsaken and this covenant will last.
4. The final Covenant which you notice they all overlap each other and the next one does not negate the previous one, will be through Mashiach, per Yeshayahu and others. It has nothing to do with virgin births or human sacrifice for remission of sin or any of that. See other answers for qualifications of Mashiach.
All the commandments in all covenants are tools, little spoons to move veritable mountains, for us to choose G-d and good and overcome destruction. We start with our little spoon, and G-d moves the mountain. All of our spoons are little compared to G-d, so pick your covenant that is right for your soul.
In summary, while the covenant of Noah doesn't include Shabbat or Circumcision, neither are forbidden according to most Rabbis, except certain Orthodox. Just as pious Chassids who love HaShem go further than what is required. Non-Jews can do that too. Noahide is the minimum.
Another example: Some Hareidi Orthodox will interpret Rambam's statement in Mishan Torah that women do not receive a reward to study Torah to mean women are prohibited from study of Torah. This is not the interpretation of most communities, and women are not prohibited from learning Torah. The same applies to non-Jews who want to do more.
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Addition: As I was reading this morning from Rav Kook about the Inner Light of Israel, here is what he had to say about redemption (salvation): Redemption an Ongoing Process:
The redemption continues. The redemption from Egypt and the complete redemption of the future are one unending action: the action of the strong hand and outstretched arm, which began in Egypt and works though all eventualities. Moses and Elijah are redeemers in a single redemption; the beginner and the ender, the opener and closer together fill the unit. The spirit of Israel hears the sound of the movements, the redemptive actions, brought about through all eventualities until the sprouting of redemption will be complete, in all its plentitude and goodness.
XXVIII - pg. 139
Every bit of righteousness helps bring this about. This is why when asked who should study Kabbalah, Rav Kook, the first Chief Rabbi of Israel, said "anyone who wants to".