I have a 2005 Triumph Sprint ST 1050 with a seemingly random starter problem. Sometimes it will turn over, but that’s it. Then the next day it works fine. The dealer has swapped the plugs over and back again and it behaves itself for a few days,then the problem returns.
This sounds as though you may have a poor connection somewhere. There are a couple on Trumpets that can attract moisture and if you’ve had technicians poking around to see what’s up they are inadvertently remaking the connection temporarily, which is why it runs for a while.
The favourite ones to check are the pick-up coil on the right side of the engine. The connection plug is just below the throttle bodies. Look at the sidestand switch with its connector just above the front sprocket cover and also the clutch switch, which has its connector is behind the headlight.
Pull the plugs apart and clean them with contact cleaner spray, apply something like white grease to seal and put back together.
We would do the same with the main loom connectors behind the speedo/clocks too as this will cover the ignition and killswitch circuits.