Specialized Top Mount levers.
I wouldn't get them, on account of no adjusters. Unless there are adjusters elsewhere in the cable run, that would make brake setup more difficult than it should be, but if they are set up properly, so there's minimal brake lever movement before the caliper arm moves, there's no reason why they shouldn't brake well.
It's good practice to bed disc pads in with a few hard stops on each brake before using them for regular riding. You can try out the top levers at the same time and see how they feel. As
@Ajax Bay said, the drop bar braking won't be affected, unless there are poorly cut outer cable ends at the levers.
Your problems with the old bike seem normal for cantilever brakes, which have always been fussy about setup, and with the required cable setup changing with pad wear.