One input, even if you seem to have solved your problem..
I had a similar failure on my T3.. The problem was linked to dirt within the starter.. I clean it and everything came back to normal..
BTW to perform the job and diassamble the strater you will have to flush the coolant .... This amazing mechanical accessibility on our T3 :hammer:
Note that in some cases the brushed has to be changed.. but Triumph builded 2 brands of starter on their T3..
Nippon denson, and for them the brush are available as spare parts
Taigene, and the brush are not available, you will have to buy a complete starter

But in that cas the best solution is to bring the item to any good shop specialised in elec and wiring and they will be able to renew it for a decent price ...