Well, yes and no - I haven't been in the industry in a while, but Snapper used to require special build engines from the manufacturers. I can't imagine they don't do that now, especially reading this article. We sold them a Subaru (Wisconsin Robin) a few years back - don't know if they still use it. I think they went with another Japanese engine for that segment of the market.
But - Snapper made Briggs come up with a little higher quality engine (ball bearings on the crank, etc.), and they have used special-built Tecumseh engines as well. Snapper ain't no slouch.
I have a John Deere rider, by the way. Paid twice as much to get a real one from a dealer. As for a Home Depot "John Deere" - they ain't really John Deeres, if you ask me. Lots of significant engineering differences. JD used to call them "Sabres", but must have bowed to the low price pressure of Home Depot and put the JD badge on them, just like the Snapper guy in the story refused to do. Good for him.
The choices on my mower were...
11-hp Briggs & Stratton recoil
12-, 13.5- or 16-hp Briggs & Stratton electric start
11-, 13-hp Honda electric start
15-hp Kohler electric start
The Honda was a good deal more expensive than the B&S of the same size. The Kohler was considerably more expensive than the Honda, though I hear that the Kohler will last insanely long. If I were a professional groundskeeper I'd probably get the Kohler.
I ended up with the 12-hp B&S electric start.
Are you sure about the ball bearings on the crank for a four stroke engine from Briggs?