Gaggia Syncrony Logic Dispenser drive operating a slow speed

Tanya has it
https://elektrotanya.com/gaggia_syncrony_compact_digital.pdf/download.html

more her but only User manual.

https://www.google.com/search?q=gaggia+syncrony+logic+service+manual

The unit requires a new grinder assembly

Check the chamber under the white plastic valve behind the brew unit, it might be blocked with coffee.

Check the site at importika.com they have a list of repair areas any one of them will have a spout in stock.

Yes the machine should not be left on for that long, it only takes a few minutes to heat up. It does not have an auto refill if the boiler is low on water.
If it heats up the water in the boiler for over an hour, it is safe to assume, that the water might at some point turn to steam, when you use it for the first time, it flushes out all the steam, and fills the boiler with water.

Lubricate the brew unit with a food safe grease this should help.
The brew unit becomes stiff and fail to load up fully this stops the water being dispensed

Sounds like the brew unit is blocked, to be sure try this, use the pre grounds button to brew a coffee, if no water comes out then it is the brew unit, remove unit and look under red o-ring, you will see a shower screen, clean shower and tube that flows to the discharge(top, you will have to remove the 2 screws up there, be careful not to lose the small plastic ball under the spring). Reassemble and try again.

I switch the machine on and the green light (number 24) flashes to indicate it is warming up but it never gets there it just keeps flashing for at least half an hour.

Try this press the button for ground coffee then when it lights up press the coffee brew button, if water flows normally then the grind on the grinder needs to be adjusted coarser, if no water then the screen on the brew unit is blocked.

Need also to find height adjustable coffee dispenser, also need measuring scoop,brush and coffee dispenser with wrench.