...or even start to say 'Sir, about before...' and have Picard stop him and say 'No need, Commander.' Something like that.
Yes, that would be appropriate as well. Probably better than a full on apology.
It would have been a nice character bit on both their parts. Demonstrating that Sisko can admit the wrong and that Picard doesn't hold a grudge.
Really I never got any impression from that final scene from Sisko and Picard on a personal level between the two. It was obvious that Sisko was no longer disenchanted with Starfleet and found a purpose, but the rest really is speculation.