Yes, the Last Shot rule idea works against a central design feature in Dogs, which is that you always have the option to Give instead of taking Fallout. That’s really key to making the premise of the game work!
Having said that, don’t forget that Dogs’ rule for GMs is “say yes or roll the dice”. If a “can I hurt him?” follow up conflict sounds uninteresting (and it probably is!), the GM should just say, “yes, you shoot him,” after the conflict ends. If there’s a good reason you couldn’t… then THAT might be an interesting conflict, but generally, I’d just say it’s good and done - he’s in front of you, you pull the trigger… we all know how that ends.
Edit: And if it’s a PC vs. PC conflict, don’t forget that you can apply “say yes or roll the dice” as a player, too. Sometimes, “I just stand there. Do you really shoot me, just like that?” might be just the right thing.