First off, most all of these air cooled engines have other uses such as lawnmowers. Winco used to use Briggs engines as did Generac years ago. As far as I know, Kohler has always used their own air cooled engines. Briggs and Kohler engines have of course been built for mowers, pumps etc. I believe Generac has also sold engines to OEM's for mowers and such.
Second, vertical crankshaft engines have been built for decades. At first, I have to admit, I was sceptical of hanging an alternator on the bottom of one. The vertical crank does allow for a much smaller footprint of the generator. Also, considering the relatively low hours these engines normally get, it's not likely that a seal will wear out any time soon. Even if the seal does leak, with regular maintenance, the leak should be noticed and repaired before it could ever cause any serious damage. Kohler has been building their smaller, air cooled, vertical generators for several years now and I have installed a bunch of them with no issues.
I'm pretty sure that the larger Briggs generators are horizontal shaft.