Part of the problem is the aiming point you use on final is not the landing point, but the flare point.
Everyone is told to aim at the numbers, for example, and adjust if they seem to move up or down in the windshield. Thats fine if you don't flare, but the roundout and flare will use up about another 500ft before the wheels touch, more if you're above target airspeed.
Teaching to land a third of the way into the runway, with a powered approach, is in case the engine quits. That way, even if you land short, you're still on the runway. Not really a good idea, as even a perfect approach still leaves you with at most only 2/3 usable runway.
If the wheels are not firmly down BEFORE the first 1/3 of the runway is behind you, go around!