Order vs. Randomness

[Alex K]

Since orderly systems cannot arise from purely random processes, nor out of nothing, it must come from an underlying pre-existing order.

The following can be said: the perceived arrow of time in our universe comes from the fact that things macroscopically move from lower to higher probability states. Within a perfectly thermalised heat bath in a static universe, there wouldn't be an arrow of time as we know it. This is a separate question from whether quantum measurements are true random events.

And, in my mind, the concepts "order" and "design" are inseparably fused together

That's just prejudice

with the idea of an intelligent designer because order is an aspect of intelligence, not of randomness.

You are merely prejudiced to assume such a complicated contrivance such as an intelligent designer without realising how contrived it is and how little it solves, because you are as most of us, part of a culture that sees it as some kind of default explanation. If you think about it, it doesn't even solve your problem of order and randomness! Postulating an intelligent designer to explain "order in the universe" necessitates a further explanation why there is an intelligence to start with - you have gained nothing but introduced an gigantic unnecessary complication!