I make straps with English Bridle leather, too. This is my process:
I use a mixture of beeswax and candelillia wax (80/20). My process goes like this: bevel edges, sand if necessary, rub plenty of wax over the edge (it should completely cover the surface), then burnish on drill press, then burnish again with canvas. It's quite a lot of work, but by going through all these steps I achieve a burnished edge that doesn't fray and won't unravel over time.
If you want dyed edges, you need to sand and burnish with saddle soap before dyeing. Then after applying dyes you can add wax and burnish again. This is basically Bob Park's method which is excellent. I don't dye my edges because it would put my products at too high a price point, but if I were to, I'd take this route.
It took me a long time to get the process that works for me. After figuring out how to properly burnish, I realized why so many manufacturers just use edge paint. It saves so much time.
But what silverwingit said is true. No burnished edge retain its smoothness for years and years on straps that go through a lot of flexing. I think that's ok. Acrylic edge finish doesn't last forever either as it'll chip off with long wear. At least burnished edges can be easily re-burnished.
Hope you nail it. There's not much in leatherwork that is as satisfying as a beautifully burnished edge.