Australian runner finds fish (spotted handfish - Brachionichthys hirsutus) thought to be locally extinct on beach during her run

How is a dead fish proof the fish isn't extinct?

Because if it was still physically intact, then it died very recently. Therefore, the species couldn't have been extinct since 2005 as the post says was previously believed.

Plot twist: it was the last of it's kind and died a miserable death

Looks a lot like a frog fish.

Where was this runner, to notice a fish while running, let alone ID it as rare?

I don't think I have ever noticed a fish while running, not even an ordinary stock trout in a creek or pond.