If your preflop raiser was on your right instead of your left then your strategic concept plays out more along the way you are thinking, but your line of thinking is still a little off here. When UTG bets out on the flop the field is faced with calling one bet and their pot odds will justify many drawing hands to call. When it gets back to you in the SB and you raise, the only one you're facing with a cold call is the BB. It just so happens that he had one of the hands you wanted to fold, but you can't know that and a raise only faces the rest of the field with another single bet to call. You are thinking of facing the field with 2 bets and the right way to do that is to bet out and hope UTG raises right behind you.

Regardless, middle set is a hand that you want to drag people along with you and not drive them out on the flop. Get them to put in those bets on the flop while you can. That's the reason for a check/raise here - so that you can let UTG bet it out for you and have callers all the way around. That way when you pump the pot with another bet everyone is getting odds to call again. Other posters have advocated betting out but that relies on enough of a read on your opponents that someone will raise your initial bet. If you don't think you'll get raised then a C/R could be right here so that you can get an extra bet in. If you assume a raise after your initial bet then you could be looking at 3SB in the pot per caller, however the number of callers may go down.

You WANT the weak 3,4, and 5-out draws along for the ride here tossing their chips into the pot for you to take down with your set. The odds of their gutshots hitting are so low that driving them out is a mistake on your part. You actually have a better chance of making your full house than a single gutshot does of completing his straight and with a non-paired card on the turn your number of outs goes up yet again.