Interesting idea, but back when I needed to enter such I use DEC's compose.exe (while it still worked) and then Allchars:
https://sourceforge.net/projects/allchars/
or used a stylus and the drawing input in TeXview.app on my NeXT Cube or https://www.inftyproject.org/en/software.html --- since then, didn't Microsoft's Journal and various other stylus note-taking apps get math support?
These days I just press the touch keyboard icon in the task bar and use that to get special characters.
A further concern is one has to be entering things more than just as a lineal stream --- superscripts and subscripts need to be entered into, things need to be put over or under a fraction bar, &c. --- do the hard-wired buttons actually accelerate things that much?
Why not just use a programmable keypad such as a Streamdeck?
WinCompose [0] is the spiritual successor to AllChars.
I love the idea of a physical device (esp. with buttons) but I get that it is not always practical. I wonder how much work it would take to add math symbols to something like Touch Portal. I haven't used it but read that many people do for controlling various software, including DAWs (which can be latency-sensitive).