I would think that a pulling lab will still pull wearing that harness (with no additional training). Harnesses that attach to a leash on the back of the dog actually encourage a dog to pull. Think of sled dog harnesses.
Front clip harnesses, such as Gentle Leader's Easy Walk and the Sense-ation/Sense-ible harnesses, are different and work much in the same way as a head halter (Gentle Leader, Halti). As the dog starts to forge ahead, gentle leash pressure will turn the dog's entire body (instead of just his head) away from what they're pulling towards, giving you both a slack leash and chance to reward that slack leash.
I would still have to have a regular collar too though. First of all I would need to have a hold on him in the elevator. Yes, I live in NYC in a building with 400 tenants and other dogs. Secondly, when walking the NYC streets I am not sure how well he would do walking behind me. I currently have a prong collar and Jax is good with correction until he sees another dog...then he wiggles himself into a frenzy and jumps up sometimes. Not with all dogs (especially smaller dogs) but if he sees another Lab...LOL