The paint stick is relatively soft wood, but there's a decent chance it'll still do. If anything, make a few more cuts into it. Don't be afraid to cut DEEP into the wood, then draw the cutting edge through it, from heel to tip. Then strop again & check your edge.
The 'shine' on your edge may not necessarily be a burr in itself. Might just be a microbevel that's been polished by the stropping. If you're not able to feel a burr with your fingernail, another way to test for it's presence is to slice a piece of light paper (such as a telephone book page). Make 3 or 4 cuts, along the full length of the cutting edge. If the edge catches or snags midway through the cut, that's a good sign you have a burr. If, on the other hand, you can repeatedly slice cleanly & smoothly from heel to tip without snagging in the paper, chances are good the burr's gone.
Sometimes I've been able to see a burr 'scrape' little bits of leather from a strop while stropping. If the strop has a layer of colored compound on it, it's all the more visible. When stropping the side with the burr, the burr will 'collect' the compound, much as if you'd scratched the strop with your fingernails and collected compound underneath the nails.