This is an interesting way to view randomness - what do you think of “the mechanisms for these matters are so complex and variety influenced by many subtle factors, that it is beyond our current capabilities to deal with such in a scientifically meaningful manner”.
The only approach currently available is via probabilities and statistics.
That is called hidden variable theory and it was disproven. Einstein and others proposed this because they had a hard time with what quantum physics was telling us. What you and these suggested is what scientists would have liked to believe – that they just had to keep uncovering these complexities we haven’t figured out yet. But Bell figured out how to test the idea and the tests showed this was wrong. There are no hidden variables, and there really are random factors incorporated into the laws of nature.
But even though these random factors are everywhere, it doesn’t mean everything is random. Events have many many different causes. So along with these random factors are also a lot good reasons why things happened the way they did. I have suggested these random factors are a window through which God can participate in events. But it is very narrow window. An all knowing God can use it to do miracles in our lives, but it far too narrow for some non-physical puppet master (mental soul) to direct our bodies from outside the physical laws of nature.
I am responding to outlooks regarding mechanisms for biochemistry (and ultimately biology). Complex mechanisms in biology and biochemistry involve numerous interactions and processes, which might appear random at a glance. However, most of these mechanisms are far from random. They are governed by intricate and well-regulated biochemical pathways, molecular interactions, and genetic codes.
Other lineages had different starting points and different environmental pressures. Feathers are made of beta-keratin proteins, which mammals don’t produce, so mammals couldn’t evolve feathers without first evolving the component proteins, which is unlikely since they already have other proteins which perform a similar role. Also, if (as seems likely) feathers first evolved for thermoregulation, mammals already have fur so there’s no selection pressure for them to evolve a different method of thermoregulation.
Statistics (or rather probability) is definitely part of the answer - there are a staggeringly huge number of possible genomes, and the probability of two genomes that have been diverging for millions of years converging to the point where they both even contain one identical gene, let alone enough identical genes to produce the same feature such as feathers, is even more unlikely than a monkey with a keyboard producing an exact copy of Hamlet.
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