Whatever shifters you get you need to make sure it matches your derailleurs. They need to match Shimano 10 speed road on the rear and Shimano road double on the front. MTB and road lost their rear compatibility they had up til 9-speed. First the MTB changed at 10 speed and then the road changed at 11 speed so the newer ones are neither compatible with each other or anything old. The fronts were never compatible.

So far as I know the Tiagra 4600 flat bar set is the only one from Shimano itself. There was a Dura Ace 2x10 road flat bar set but it was a long time ago now and not common so good luck to find one.

If you want thumbies, microSHIFT -The best control system seems like the right one.