So I've only been on a road bike for a week now and one odd thing I've noticed is that the headset bearing has absolutely no tension on it. When I pick the bike up, the front wheel goes completely limp and flops to one side.

Now I would've just accepted this as normal but I happened to watch a coworker of mine walk his bike through the hallway yesterday with just his hand on the seat (high end carbon bike, didnt catch the brand). The bike maintained a perfectly straight line.

Out of curiosity I tried this in my driveway last night and the front wheel was all kinds of wobbly. So this has me wondering if a front wheel/fork should ideally be completely free moving, or if through adjustment or replacement of the headset should actually have some tension.

Just FYI, I have 23C tires on the bike which I suppose could contribute to this (vs a wider tire).