Yes I think a proper three bet on the turn would fold out better hands but not a min 3-bet.

(In fact just a few posts down is a hand hypermagachi posted where he had trips on the flop (slightly different scenario but he wanted to know the best play when the dude bet into him. (He had Aq on xAA flop.)
I wrote that if we just call then the best we can hope to get (from hands we beat) is a 1/3 pot crying call with a value bet on the river and that I often elect to min raise.

My exact response was..

"Or higher I like to think then min-raise, try to make them think I have a hand like QQ/KK and I'm just 'checking' to see if they really have an ace, then they sometimes try bluff push. "

Which it seems like is exactly what happened in your hand, the minraise was enable to induce a bluff shove from A5.

(Again the nature of the board JKKJ & you being pfr is what indicates the possible real strength in the min re-raise... on a 6886 board the turn min re-raise (if he had an 8) would fold out all hands he beats - higher pairs - that he could get value out of on the river - and therefore a 3bet here is more likely to be a bluff/weakness.)

Again not really sure of any of this just my opinion.