Digital demodulator (SDR) for receiving legacy analog modulated wave
Sure, an SDR can transmit and receive conventional, analog AM. I've done it many times as an introductory exercise in my courses.
A quadrature signal $s_{\text{QAM}}$ is made up of two signals, $s_I(t)$ and $s_Q(t)$, that are transmitted over the same bandwidth and modulated in quadrature (i.e. with carriers phase-shifted by 90 degrees):
$$s_{\text{QAM}}(t) = s_I(t)\cos(2\pi f_c t) + s_Q(t)\sin(2\pi f_c t)$$
There is no restriction on the shape of these signals, beyond the physical limits of the hardware or the channel (in bandwidth or in power). In particular, they can be a DSB-LC AM signal!
$$s_{\text{QAM}}(t) = (A_m + m(t))\cos(2\pi f_c t) + 0\sin(2\pi f_c t)$$
The regular downcoversion and filtering process applied to $s_{\text{QAM}}$ will produce $A_m + m(t)$. You can also apply a traditional envelope detector. That's the beauty of SDR: it does not care about the meaning or structure of the in-phase and quadrature signals; you define that in software.
To conclude, note that not only it is indeed possible to transmit/receive AM with an SDR; you can actually transmit/receive two AM signals on the same bandwidth!
$$s_{\text{QAM}}(t) = (A_{m1} + m_1(t))\cos(2\pi f_c t) + (A_{m2} + m_2(t))\sin(2\pi f_c t)$$
You select one of the IQ branches and ignore the other, and then you can listen to one &quot;station&quot; or the other. Of course, this assumes that you are transmitting your own quadrature AM signals (for example, over one of the ISM bands).