I am certain there are great use-cases for this, but it seems the discussions imply that "change your FROM, save a ton of disk space" but there is more to that story.
The requests would leave the machine, but it seemed none of the handlers were firing with the result. I thought, ok, perhaps I've made a configuration mistake or the target website is acting funny.
After 3 hours of inserting every kind of debugging print and log statement I could think of, I decided to sanity check the setup using debian:wheezy. As my post above implied, it worked immediately and flawlessly.
Thus: it may work great for you, but ensure their libraries and binaries are a good fit for your situation.
I did see quite a few other comments discussing Python, Ruby and other not-easily-statically-compilable stacks may not be a good fit for the system the article was proposing, so this may not strictly be Alpine's fault. It's just when I see a discussion of the great savings from using Alpine, I feel it's important to remind folks there is no free lunch.