Your perfectly single-sideband suppressed-carrier modulated sinusoid certainly has a phase which can be measured. However, what you cannot tell is what the contributions of that measured phase from the audio input and the RF oscillator were.
There is another form of single-sideband modulation, in which not only one sideband but also the carrier component is transmitted. This provides a reference which can be used to synchronize the receive LO to the transmit one - normally done to insure exact tuning, but it would also give you the ability to recover the original audio phase.
It is also quite possible, especially with modern DSP gear, to transmit two separate audio channels, one on each side band. This is commonly called independent sideband modulation (ISB).
Many spread spectrum implementations are DSP based and capable of receiving multiple channels at once - GPS being a good example.
If we go back to basic theory, we have a carrier signal of the form :-
\$E_c\cos\phi_c\$
... and a sinusoidal modulation signal of the form ...
\$E_m\cos(\omega_mt)\$
and if we let the frequency deviation be proportional to the modulation amplitude, so
\$\Delta\omega\propto E_m\$
the instantaneous frequency is given by ->
\$\dot{\phi_c}=\omega_c+\Delta\omega.\cos(\omega_mt)\$
Integrating this to get the instantaneous phase ->
\$\phi_c=\omega_ct+\dfrac{\Delta\omega}{\omega_m}\sin(\omega_mt)\$
So the modulated output is ->
\$E_c\cos\Big[\omega_ct+\dfrac{\Delta\omega}{\omega_m}\sin(\omega_mt)\Big]\$
As you say, the modulation index is dependent upon \$\omega_m\$ so the relative amplitudes of the spectral components will vary with \$\omega_m\$, but the modulation index is also a measure of the peak phase deviation, so if you want the spectral amplitudes to be independent of \$\omega_m\$ you must have \$\omega_m\propto \Delta_\omega \propto E_m\$, ie phase modulation.
One technique of producing phase modulation is to use a frequency modulator with pre-emphasis of the modulating signal to get the amplitude proportional to the frequency.
Best Answer
Amplitude of the signal is irrelevant once you have defined the modulation index. Use this table as an easier guide but remember it applies only to sinewaves as the modulating waveform: -
For a mod index of 2.0, your carrier will appear to be 22% of what it was unmodulated and there will be sidebands of amplitude 58%, 35%, 13% and 3% of the original carrier amplitude. The first sideband occurs at a distance equivalent to the modulating frequency away from the original carrier. 2nd s/b at 2 x distance etc..