Your instructor is half right - and fixing the half he's wrong about will clean up the rest of your flare.
The old "look away from the aiming point" advice causes exactly the problem you described. A sudden shift in visual reference mid-flare disrupts your pitch control because your hands follow your eyes. So he's correct that a deliberate "look away" can destabilize things. That part checks out.
Where he's slightly off: "keep looking at the fat markers" isn't the answer either. If you stare at the touchdown zone all the way down, you lose your ability to judge sink rate and height in the last 20 feet - the markers rush under you and your brain runs out of depth cues right when you need them most.
The technique that actually works is a progressive shift. Use the aiming point to fly the path on approach. As you begin the flare, let your eyes travel further down the runway - not sideways, not randomly "away," just progressively ahead. This is what gives you the peripheral runway expansion cues that tell you how fast the ground is coming up and whether your attitude is right.
Airbus training material says it plainly - look well ahead during the flare and landing to judge aircraft position relative to the ground. The A320 cockpit geometry actually makes this easier than most types because the sight picture is so clean.
Your old technique was probably too abrupt. Your instructor smoothed out the scan and that's why your flare improved. But the real target isn't "stare at the markers." It's a gradual transition from aiming point to runway ahead as you enter the flare. Once that becomes habit, the rest sorts itself out.