Linking Jesus to the creation assumes that God created the universe, which is asserted in other parts of the Old Testament besides the first 3 chapters of Genesis.

I agree that John 1 was influenced by Genesis 1. The main issue is whether any crucial truths of John 1 rely on Genesis 1. So in John 1:4-5 —

In Him was life; and the life was the light of men. And the light shines in the darkness, and the darkness did not comprehend it

— we see, as you point out, a parallel to the creation of light in Genesis 1. But in Genesis 1 the light is created by God; in John 1 the light is the life of God. I don’t think John meant for his readers to reinterpret the creation narrative more pantheistically based on this lyrical passage he has written. John says that "the darkness did not comprehend" the light, implying a metaphorical interpretation of light (life, truth, etc.) and darkness (sin, ignorance, etc.) that certainly isn’t evident in Genesis 1. Is John suggesting that his readers interpret Genesis 1 metaphorically rather than literally? I don’t think so, since he already asserted that "all things came into being through Him." I suppose it’s just possible that John is reflecting a common understanding that, while God did create everything, the Genesis 1 story isn’t a literal account of the creation event. That would be interesting.

My sense of it is that John 1 is, as you say, a reinterpretation and contemporization of Genesis 1. His main interest isn’t to elaborate on the creation but to introduce the incarnation. The incarnation is continuous with the creation in the sense of God repeatedly intervening in the world of men through the mediation of Christ.