If you have the newer AirPods Max with USB-C, then Apple's USB-C to 3.5mm adapter can be combined with a 3.5mm-3.5mm AUX cable to make a wired connection.
Sorry for spreading false speculation. I was basing this on other experiences with USB-C headphones and just assumed that the AirPod Max would work the same way. The USB-C to 3.5mm adapter from Apple is known to be bidirectional, so I wonder why that solution doesn't work with the AirPods. Perhaps they do not put out enough (any?) power for the adapter to work?
I am still uncertain about the issue with the AirPods Max USB-C port. The Beats Solo 4 is functioning correctly, offering both analog and USB-C connectivity. However, the analog-to-USB-C converter is malfunctioning. It either lacks power from the headphone or is not bidirectional.