> 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.)
Like you said, the first exoplanet was detected in your lifetime.
We haven't even had the chance to fail yet, the Fermi paradox is not yet in play when we're considering essentially our first move. To extend an analogy from further up thread, it'd be like looking in the Mariana Trench for un-contacted human tribes after taking a quick glance around the neighborhood and deciding there's nothing else to be found anywhere else.
Or maybe we start with what we know (carbon-based), and keep our minds open to other possibilities. Like we do now.