Running a modulated oscillator on HF is a menace. They sound really bad as compared to a normal AM signal on any receiver - even a broad one (like mine).
It *used* to be illegal to run a modulated oscillator, and this was in the FCC rules. The section was eliminated, I suppose, because the FCC thought no one would ever run such a thing in this day and age, and considered the rule superfluous.
When you have incidental FM on and AM signal, the sidebands generated by the FM signal add to, or cancel the AM sidebands resulting in a signal with more energy in one sideband than the other. Furthermore, depending on the IF slope, the FM will be demodulated improperly adding more distortion. The carrier and power input of modulated oscillators can be unsteady, resulting in dropout and overmodulation.
Such a thing is only effective as an 'SBE' if the interfering sideband signals are more or less zero beat with the AM signal, -and- if the AM signal is very strong.
If you really want to make an 'SBE', don't use a modulated oscillator - use something that PULLS the oscillator to a greater or lesser degree depending on the peak modulation. The pulling should take place at a low frequency (a few cycles per second max) and could pull the oscillator several hundred cycles one way and the other.
In this way, you get the wobulation effect without the incidental FM of the actual audio.
I confess to having built a VFO to do this back in the 1970's, tying the FM input of the VFO to a quasi-peak detector detecting the AM, with about a 200ms decay time. Effective to its purpose, but not distorted. I used this VFO system with a KW input rig (a pair of 304TLs modulated by 833As). The oscillator would pull a few hundred cycles in either direction, depending on the modulation level. Most of the time the FM input was not active.
I AM NOT ADVOCATING SUCH A THING, but simply saying if one insists on trying to create an 'SBE', there are far better ways to go about it than using a modulated oscillator