I just took my '82 6.5 kW to an Onan service center (to find out where the reset button was!) and in the process learned a little about how they work. You are correct, it doesn't have a starter per se. As I understand it, it starts by way of an internal electric motor (commutator) that spins the gas motor up to starting speed. It is separate from the electric motor that makes power.
You probably already have, but if it's like mine, check the two fuse links on the front panel. The tech told me that's the first place to check. Other problems can be the solenoid , fuel delivery, plugd, basically anything that causes small motors not to run or to run poorly. Seems like I did a seach of this site using "generator" and someone had posted the troubleshooting section from the AS motorhome manual. That might help.
Another forum that might be useful is a small engine forum I found that has several posts about Onans as well as a guy on the site who is an expert with them. If you sign up and post the specifics there, it looks like the responses are very technical and useful. http://www.perr.com/forum/