> Or maybe we start with what we know (carbon-based)
But we know more than one thing. One of those things is the Fermi paradox - that the universe should statistically be full of evidence of life, and yet we struggle to find it. That may be evidence we're making the wrong assumptions.
> keep our minds open to other possibilities...
Yes, and I'd argue that means including scenarios like Jupiter's moons in our search. (As a bonus, Jupiter-style planets, being larger and far from the star, are substantially easier to find.)