It depends on how wide the runway is and yes, the C-172 can land with considerable drift on and stay on the runway.
What I like about wing-down is that you get a clue ahead of time when there's not enough rudder to keep the plane lined up. Mind you, there will be less wind at the surface.
If the wind picks up another 10 kt or so, do you know where to find another runway that's pointed into the wind?
Once you're down in a wind over 20 kt. and still on your wheels, be very careful about taxying downwind.
As for the FTO, they may be used to the prevailing winds and if you have demonstrated acceptable landings dual in similar conditions, they may have decided that you're up to the job. Students botching landings gets the attention of the CAA and raises the insurance premiums.