There 3 components to why a TT turns:
1. The moment (rotating force) applied by your body to the water AND the pull of the kite
2. Rail engagement with relationship to primarily sidecut shape (rail), and secondarily, the amount of rocker.
3. Forces from the fins via the rider induced moment, and/or the turn induced by engaging the sidecut
For #1, think about kiting along and then suddenly throwing the board sideways. This is the moment that you can apply without utilizing the sidecut/bottom shape, or the fins. This is the primary evidence for this being a distinguishable force.
For #2, a diagram of standard wakeboard sidecut vs reverse/snowboard sidecut.
So you can see that when you engage a rail that is curved, the average of that curve acts as a keel (not foiled). This is the primary turning force you can feel even when you do not try to induce a moment (by twisting your body), but rather lean back and to one side (don't lean back as much on a reverse/snowboard sidecut - you stay more centered).
And for #3, as a TT is turned by either #1 or #2, the fins engage at a higher and higher angle of attack (AOA) to the wind, generating more force until stall (spin out/ventilation). This helps reduce sideslip and actually adds to the turning forces which you can apply via your body or with rail engagement. Surboards rely heavily on this with larger fins as a means of turning. Smaller fins would be overpowered with the rear foot all the way back on the board, and thus the weight further back than you could achieve with a centered stance TT.
Drag, is not a primary means of turning. Though thinking of it as such does not hurt the mental approach to the issue at hand.