It was mainly me dreaming of having a connection to my phone. Remotely turning it off and on, seeing it's power status from the other room, LAN party theft alarm, etc. If it's not obvious, I tend to digress...
Oh dear god. I mean, that's a pretty cool idea, and maybe something that could be integrated into a separate product in the future, but right now that would mean I'd have to model a meander antenna inside the button (which I don't know how to do properly), put all the bluetooth components inside (which there is not enough space for) and write a mobile app in addition to the firmware and software I need now. Sorry, not gonna happen any time soon.
They do seem rare as hen's teeth, but they're coming from somewhere for use in the Fire TV stick, OG Chromecast, Intel Compute stick, etc. Alibaba has a handful of Micro HDMI - Micro HDMI cables listed, so that might be an option to cut down on component count by using a micro-HDMI socket on either end.
You know, seeing the rarity of all these components, it seems to me like a custom cable isn't such a bad idea after all. There are manufacturers for this kind of stuff, and they'd only have to cut a micro HDMI to HDMI cable up and add a three 0.1" pin headers to it. That's actually something I could do myself for low quantities and material cost would be quite low.
As Hamburger Eating Ironman mentioned above... the Micro HDMI - Micro HDMI may be your only choice and could be secured via hot glue to the board and then connected via a socket and/or wiring to the board.
Ah I see. The problem with that is that these connectors are super small and need to be hand-soldered, so the manual labor cost would be very high. To be suitable for low- and high-volume manufacturing alike, I need to make sure most of the assembly can be done automatically.