Sounds like the IC is what is left, or a trace to it.
Common earth point (ground) yes the frame of the bike. Other than that the battery.
Triumph does not impress with how they wire things. If you look up in the headlight area you will find several thin ground wires running back to a slightly larger wire. All ends stripped back to copper and crimped together.
Could it be a ground issue, yes. Have seen ground issues cause some real weird stuff. (could be wrong) But don't think so in this case.
IF you had odd things happening randomly and you did not touch the cluster, then yes a ground fault would make sense.
Since the problem only occurs when you push the button the problem sounds more mechanical like a wire, solder joint or bad trace. The latter being something to consider.
Keep this up, will make a electronic tech out of you.
Not knowing what would cause the IC to revert the display back to ground makes this tough.
One would have to assume removing either a voltage or ground to the IC would cause the chip to default. (is this a default setting or a reset condition)
You did the easy bit, the next step IF you are determined enough makes the last look easy.
Let me know if you want to wade into the step.