USB Type-C Alternative with fewer signal wires
If you are looking for "sleek look" then I think you are out of luck.

All these nifty compact reversible connectors are relatively new invention, and they all were designed with specific protocol in mind. Therefore your "with any sanity" reservation will be applicable to any of them. Whether you have Type-C, Lighting or magnetic charge port, sooner or later someone will try to use it as expected by its appearance.

But if you would be OK without "sleek look", then any non-polarized connector will do. Even a ubiquitous rectangular connector can be used if you get 2x5 pins and wire them properly.

UPDATE:

I have no idea how expensive manufacturing your own connector is. However my guess is that unless you are talking about huge volume (literally by millions) then manufacturing proprietary jacks and cable assemblies of the USB Type-C quality would be cost prohibitive.

I would recommend finding suitable rectangular connector and designing a shroud for it instead. This can be easily prototyped on 3D printer and can be manufactured in small volume by injection molding (relatively) cheap.

Note, that 1.27mm pitch is not really compact. There are plenty rectangular connectors going from 1mm all the way down to 0.25mm.

Finally, are you sure you are not fighting another XY-problem? Yes, the reversible connectors are cool and all that, but there are polarized compact solutions that are still easy to use and readily available.

There are circular connectors with any number of pins from 2 to 40, there are interface connectors, like Hirose ST and LX series. Even good ol' 6P6C RJ25 modular connector will do the trick.

Also, if "cool factor" is what you are after, then there are tons of dirt-cheap magnetic pogo connectors on Chinese sites, like these from HytePro company. They are magnetically polarized, so cannot be mated in wrong direction. Keep in mind that they require high level of TVS and shorts protection, though.