Definitely works! I stained/dyed some chaps last weekend with Minwax oil stain in English Chestnut. The chaps were a buckskin tan/grey, and the English chestnut turned out a beautiful deep reddish-brown.
It did NOT stiffen the leather at all, quite the opposite. I'll be conditioning soon though, after I'm confident they have gassed off enough.
A couple caveats....the "paint thinner" smell takes a while to gas off, so I would leave whatever piece you stain/dye OUTDOORS for 3-4 days minimum. Maybe more, depending on your temps and airflow conditions. Since I did the 2 halves on different days, they have gassed off at a different rate...3 days on the 1st half, and the smell is just about totally gone, but the other half still had a bit of the smell. I expect the 2nd half will be ready to go tomorrow.
When I buffed it out, the excess stain was sticky as hell...so I used ISO alcohol on the buffing rag and buffed away. Removed the sticky quite well. Also removed just a trace of color on the raised areas so gave it a little bit of the antiqued/worn in look (which I really like)..very, very happy with how they turned out, and I'd use oil stain again in a heartbeat.