Good point – I hadn't considered the phone headset.
Both mic and headphones work – input and stereo output – without manipulation in a standard Devuan (Debian without systemd) laptop (HP hardware, with Intel integrated sound and a single 3,5 mm audio jack). Of course, this is not directly comparable with the OP's hardware, but it does indicate that Linux can handle a TRRS headset, so the solution probably lies in the direction of quirks/options for Asus hardware.
Addendum: Just tried on Madame's junky Lenovo with Mint, also Intel – works as expected there, too. Both my test cases have xfce Desktop, but I wouldn't expect that to be significant in this case.
What is the audio mixer showing (access depends on Desktop Environment)? In Mint, the speakers/headphones are on a separate tab in the mixer to the mic (output and input, respectively).