I'm not from a D&D background, so I may not "get" it, but modifying the target score beforehand and then comparing the result of the dice-roll seems more straightforward to me...
I'm not particularly invested either way. I don't mind the reduction in target number as a way of dealing it, but I wouldn't care if they left the DC the same and added the modifiers after, either.
That's probably better, because there actually is a difference in effect, though. If a check has enough modifiers to it, the "target" could be be reduced as far down as 1, in which case you could pass it even with a natural 1, which by the rules is an automatic failure.
I actually encountered one check with a target of 1, and that was where I rolled my first natural 1 (in dialogue), and passed the check. It was the tutorial for using the Illithid tadpole, so the check was easy on purpose, but it still exposed a potential weakness.