Sorry it took me a few days to get back to you Tim. It could be the pump, but its pretty unlikely.
Mark's probably right, somewhere along the way, your fuel line is gelled. I run standard gas in my Honda commercial engines because I've never had an issue with the gas gelling in them. I run non-oxy in all the others (ryobi, stihl, even deere)
It wouldn't be the electric start, it pretty much sits idle once you're running. If your not sure, you can pop a terminal off the battery once its running. (BE CAREFUL! DC current is no joke!)
Did you try the Seafoam thing? If you took it into a small engine repair shop, that's probably where they'd start. They'd call it "cleaning your carb," and charge you $50 or more for it. Another option is do the same process with the stabilizer Mark recommended mixed 50/50 with fresh gas.
Could also be you're not getting a good spark. Check the plug for gap and spark. You might even need to sand the contacts a bit. If you're still not sure, a spark tester is like $10 and a plug is like $5.