Actually evolution makes MORE demands on the evidence, and therefore is MORE prone to possible falsification. If a gene (or a region of a gene) is relatively more highly conserved, then there should be some functional explanation for that. As you note the purely functional differences could be explicable without evolution. But common descent requires that genealogical pattern ALSO exist, AT THE SAME TIME.
In other words, even if a gene is relatively more highly conserved (as compared to other genes) across, say, humans, birds and fish, it STILL must be more similar (or at the very least, not less similar) in humans and birds versus either as compared to fish.
Then there are even MORE patterns that must SIMULTANEOUSLY apply in the same gene sequences if evolution is true. For instance many genes can clearly be grouped into families. It is now uncontroversial that this pattern often results from gene duplication events. An extra copy of a gene appears in the genome, and can then "evolve" into a new gene with a different (if typically similar) function than the "parent" gene.
Now it's required that the sequence data correspond BOTH, and SIMULTANEOUSLY, to the genealogical relationships when the same gene is compared across variously related species, AND when different genes in the same gene "family" are compared within any one species. And MORE THAN THAT, the various genes in a given gene family must (absent some functional difference) show the same degrees of relationship one to another when compared in ANY and EVERY species that contain them all.
The is NO nonevolutionary reason that all of these different patterns be encoded into sequence data. ONLY common descent explains ALL the data.
Whether a gene is 'highly conserved' or not is an artifact of the initial assumption, that of naturalism and therefore common descent.
That there are functional reasons for similarities is trivially obvious, without an appeal to 'conservation'.
*Any* genealogical 'pattern' is acceptable, hence 'birds and crocodiles' or 'humans, chickens and puffer fish' is perfectly fine, though patently absurd.
A common designer explains the data much better, without appeal to absurd statements like 'birds and crocodiles share a common ancestor'. Evolution fails based on the extreme credulity necessary to accept the conclusions that are the result of the primary premise (that of naturalism).