Plant into existing gutters as a way to filter rain water...

While looking at the plants sprouting in papaw gutters my twenty year old son came up with the idea of purposely filling gutters with perlite potting mix and planting into it, in order to filter the collected rain water.
I have not been able to find such a thing, so I'm turning to Permies for input.
I imagine filling one of the fabric tubes that go over corragated drain pipe with potting mix combined with alfalfa seeds.
Could this work?

Never seen such a thing, but it sounds awesome. My biggest concern would be weight. They use all kinds of interesting things like cork and expanded shale to make light-weight growing media to get around such limitations. You can also look at pond filter media as a growing media, which has been used in vertical gardens. The pond filter media is a rigid material and won't wash away, which would be a nice advantage.

I love the idea of a roof lined with plants... buuut around here, on most houses, gutters have a useful job to do! Namely, channeling all that rainwater to a downspout, so that it can leave the roof en route to wherever it is we'd like it to go.

If the gutters were filled with grow media, roots, and a bit of accumulated organic matter, in short order they would not be able to keep up with the flow, and the water would be back to running over the edge and perhaps ending up somewhere we would rather not have it...

I think a filter is best left as a separate unit, but I don't see any reason you couldn't build a filter as described using old scrap gutters, receiving the water after it has left the roof, in a location that an overflow won't be a problem. I'd think to avoid it just turning into a swampy morass you might want some sort of prefilter/screen to keep out the larger detritus, though.