The Yismach Moshe section of the prayer leads to VeShamru, which is taken from Exodus 31:16 to 17. (Our discussion will involve a few more sentences, Exodus 31:13 to 17.) Rabbi Avudirham discusses this section of the prayer. He notes (in my own translation):
They instituted the recital of “Moshe rejoiced” on Shabbos because of
the first chapter of Tractate Shabbos (10b) [quoting Ex. 31:13]: “To
make known that I HaShem have sanctified you.” The Holy One Blessed Be
He said to Moshe, “I have a good gift in my treasury, called
‘Shabbos,’ and I want to give it to Israel. Go let them know.” That’s
why they instituted [this section -- to make known] that Moshe
rejoiced in this good gift of Shabbos that was put into his hand.
The gemara that R’ Avudirham mentions (Shabbos 10b, like the similar statement at Beitza 16a2) discusses the proper way to give gifts. It refers to a couple of consecutive sentences of Exodus Ch. 31, including 31:13 which R’ Avudirham quotes in turn, to illustrate that when you give a gift you should let the recipient know, just as Hashem caused Moshe's face to shine when giving the Luchois to him and to the Jewish people to make known that HaShem has sanctified us. For example, when feeding a friend’s child in the absence of the parents, one might put cosmetics on his face so that his parents will ask him to explain what happened (a strange example to us, but apparently a familiar one to them).
The flow of thought in Ex. 31 and the gemara are somewhat like that in these two paragraphs of the siddur. Moshe rejoiced in the gift of his portion, and You [haShem] placed a crown of splendor on Moshe’s head [the radiance of his face] and gave him the tables. So (assuming that the gemara predates this prayer) I would suggest that the author of the prayer was resonating this gemara obliquely.
In other words, despite the placement of the words Vechein Kosuv Besoirasecha, the passage is not cited to prove that Shabbos is commanded on the Luchois. It is rather meant to connect us to this sublime thought of the sages. The radiance of Moshe’s face reminded his generation that Shabbos is a beautiful gift to the Jewish people, and by recalling this exposition we are ourselves reminded that Shabbos is a beautiful gift to the Jewish people. The siddur then proceeds to cite a commandment to observe Shabbos from this same context, Exodus Ch. 31.