Can single-sideband (SSB) be used with PSK, QAM or APSK?
My question is, what compels us to transmit the frequency-shifted version of the negative frequency part?

Indeed, nothing compels us to do so. Transmitting the same (baseband) signal twice is inefficient, it unnecessarily doubles the use of frequency space.

why isn't this done for digital modulation methods like PSK, QAM or APSK?

What makes you think it is not done ? The modulation does not matter so you can add OFDM to the list.

Actually SSB is a bit "old hat". What is done with modern modulation techniques like OFDM, OFDMA is that a baseband signal is created using digital signal processing. This digital signal is then converted into an analog signal by an AD converter (ADC). OK, I lied, two of those actually.
That gives us two analog signals. These are called quadrature signals. The essence of this signal (consisting of an I and a Q component) is that it has negative and positive frequencies !

Negative frequencies ? Huh, you must be joking !

No I'm not, a negative frequency simply relates to the phase of a signal. If you have only one signal, you cannot determine the phase because a phase is always relative. Now I hope you see why we need two signals because then I can "play" with the phase and make negative frequencies !

This quadrature signal can then be mixed (in a quadrature mixer !) with an RF carrier and transmitted.

If we would start with a baseband signal of 10 MHz wide it would exist between -5 MHz and + 5 MHz and after mixing it with a carrier of 1000 MHz it would occupy 995 MHz to 1005 MHz.

That's what is used in GSM, UMTS, CDMA, Wifi, nearly all modern digital communication standards use this.