I'm sure the 'weird' waveform is directly attributable to the design of the generator head. It's a very basic, simple design with straight rotor slots and certainly no fancy engineering. Brush type self-excited heads are about as dirt simple as you can get. Whether electronics really care much about it is open for debate. I'd be willing to guess that few things in your house are going to have an issue with the power quality.
The upside is that the ST head is rugged and cheap. About the only thing that can go wrong besides needing a set of brushes now and again is the bearings, which are just standard ball bearings. The stock units are likely low quality compared to some top notch parts (which are still made in China anyway). I've never heard of anyone burning up an ST head.
The downside is that a 'real' head, such as the Marathon Lima MAC series, which has skewed rotor poles, separate brushless exciter built onto the rotor, etc. is going to run you about $1500 for a 5kW size. Yikes! :-o I never got back to the salesman to see if there were 'quantity' discounts, etc. or other way to get it cheaper. But, that would be the cat's meow for a gen head.
Chris