The comparison part between +6 Azu and +6 Shiftry under the sun is to show him that, the context he is talking about is not appropriate. When you already have that on the field, there's nothing to talk about anymore. Your sweeper has perfect setup already. By the way, I have never liked banded Scizor. He has so much staying power with his typing and Roost, his ability to help his teamate with a free-switch on unfavored matchup with U-Turn, his priority with Bullet Punch to finish off weaken Scarfed Sweeper that it's not worth putting a Band on him to try hit and run. But it's just my playstyle, that's all.
As for Shiftry, I might want to hear about some "very fast scarfers" who can outspeed him under the sun. I invest 180 EVs in Speed, at level 100 he has 265 Speed. Double under the sun makes it 530. Choice Scarf only gives you 1.5x. Latios 252 EVs Speed Scarfed only reached 525. There are only a few Pokemons with Scarf can actually outspeed him under the sun with 180 Spe EVs, namely Weavile, Azelf, Alakazam, Crobat, Jolteon (all 252 Spe EVs). But unfortunately, rarely anyone runs those Pokemons with Scarfs.
Anyway, back to the BellyAzu matter. People run Scizor, Ferrothorn, Ditto, and now Talonflame all the time. They don't really use them for checking on a BellyAzu in particular. Ditto is a hard counter. Scizor can check many other threats along with BellyAzu. Talonflame is a fine sweeper with priorities. Ferrothorn is one hell of a lead. They serve many different purposes, which include checking on that Azu if the enemy has one.
I don't say that Azumarill does not have power. It clearly does. But BellyAzu is too fragile and unreliable as a sweeper imo. You are relying on a boosted at a cost of 50% of your HP, hoping to survive with your Sitrus Berry. If that's all then I might find it useful, but again, relying once more on a priority (Aqua Jet) with a base speed of 50. So many thing that you have to worry about, so many opponents you have to eliminate or check on in order to have this BellyAzu goes to work.