Your late model GRiSO has some weird wiring[You must be registered and logged in to see this link.]
It has 2 Start relays, the normal Start Relay (2) that the ECU holds On and the Startup Maintenance Relay (3), that you hold On with your finger on the button.
The speedo is going blank indicating that the Voltage is collapsing either the battery is dropping right off or you have Startus Interruptus.
A new battery will probably take care of it, if not a minor re-wire will.
If you were to watch the number plate light on the bike, it's wired to the same source of 12 Volts, you will probably see it dip down in brightness just before the relay drops out.
As Kiwi Dave says clean up the battery and smear some Vaseline on the connections just in case its not just dirty terminals.
I suspect when you just jab the starter the ECU is detecting a low battery and dropping out the start relay
When you hold your finger on the button the ECU is still dropping out the start relay but the Start Maintenance Relay (held by the start button) is keeping the starter going.