I'm not well versed in the world of port design, but what comes to mind is a little shallow magnetic nub with a couple of contacts on it. Easy to clean, impossible to break by accidental torquing, not deep enough to get stuff stuck in it.
The cool thing is that whatever the new design is, making adapters for 1.44mm to the new thing is dirt cheap since it's still just an analog connection.
Texas Instruments explains the scheme in a document, "tidub66.pdf."
I know USB-C connectors aren't quite what you had in mind, but they are smaller and more rugged than the 3.5mm connector (in my opinion).
Downside is it is not widely supported, but it is possible. The host device needs to support a way to detect the analog USB-C audio cable (recognized through four shorted pins) and then to split off the analog audio signals before they hit the internal USB transceiver chip.
Probably need to start shopping again soon cause updates stop in June.
Right now, since I buy audio files anyways, I've mostly been relying on a tiny audio player that can also act as a Bluetooth receiver for my phone.
Would you share more? I've never had an issue with a USB-C cable. Helps to buy well constructed ones with legit specs.
And when usb-c phones disconnect just a little, usually the phone will pause the music completely and disconnect, whereas the aux headphones will just keep playing. So if the connection isn't perfect, the usb-c cable becomes unlistenable because I can't walk 20 steps without it pausing.
edit: I've tried many cables and dongles, so if you don't have this problem, it might be just that I move around more? Biggest problem for me is commutes and walking around.
Compared to 3.5mm where the frustrations I remember were usually limited to sometimes getting a bit of a crackle or one of the audio channels dropping out and worst case scenario you just unplugged it and put it back in and it usually worked. With USBC you have to wait to see.