Interesting. Lee makes what, seven or more quite different presses? So addressing "a Lee press" opens a large question. At least five are quite strong aluminum alloy so they are light but the Classic Cast and Classic Turret presses have very substantual cast iron bodies and are capabile of loading .50 BMG. I wonder which Lee press you felt was "flimsy" but never broke?
Easy. It was back when I was first learning how to handload, on borrowed equipment. A freind loaned me a lee press that he no longer used. I dont know the name of the particular model, as I didn't buy it or purhase parts for it. It was an aluminum press, the one that comes in lee's intro-level kit.
The press loaded pistol cartriges flawlessly, but I noticed it flexing when I loaded 308 on it. It never broke, but there were times when I was resizing range brass that I worried that it would. The handle on this particular press would loosen up quite often, far more than any other press I've used. My freind later told me that he had once broken the handle out of it a few years before when loading up a few hundred rounds of 20 vartarg. Lee of course fixed it.
The cheap lee presses are an adequate press for the money. It will do the job it was intended to do, but I feel somone that wants to load on a regular basis would be better served by spending a little extra money and getting a strong steel press.