<p>Quite a few methods exist for passing data into a Flex binary from a Rails application.  Right now, I'm using the old e4x resultFormat with a xml.erb template.  I've done AMF before, but I feel like inlining parameters into the embed itself is a better solution because you don't have to wait for the browser to load a swf binary and the binary to make a web request.  </p>

<p>Taking all of that into account, what is the best practice for rendering a Flex widget with a Rails back-end these days?</p>

Sounds to me like you shouldn't use Flex at all. Flex is great for writing applications that load and display data, but is a huge overhead if you just want to display charts. If you're worried about delivering the SWF and the data in two separate responses, you should be worrying about the 200-300 K overhead of the Flex framework.
Go with something more lightweight like
Flot
and deliver the JavaScript code that generates the charts along with the data. I don't think Flex is a suitable platform for what you want.