Why is the sorites problem a "paradox"? Isn't it fundamentally a problem of definition?

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The sorites problem is a

The sorites problem is a paradox for the reason that any problem is a paradox: it's an argument that leads from apparently true premises to an apparently false conclusion by means of apparently valid inferences.

I don't think it's fundamentally a problem of definition, because the concepts that generate sorites paradoxes would be useless to us if they were redefined precisely enough to avoid sorites paradoxes. Take the concept tall man. In order to make that concept immune to the sorites, we'd have to define it in terms that are precise to no more than 1 millimeter of height, because a sorites argument for tall man exists that involves men who differ in height by only 1 millimeter. But defining a tall man as (say) a man at least 1850 millimeters in height would mean that in many cases we couldn't tell whether a man is tall without measuring his height in millimeters. Given the impracticality of taking such precise measurements in the typical case, we'd likely stop classifying men as "tall" and "not tall" and start classifying them, instead, only as "clearly tall" and "clearly not tall," i.e., clearly on one side of the line or the other. But then a sorites argument, couched in terms of millimeters, arises for the concept clearly tall man, forcing us to define that concept in terms that are precise to no more than 1 millimeter. The same cycle then begins again.

So even if we make our peace with the arbitrariness that would attach to any precise definition of tall man -- why 1850 mm rather than 1875 mm? -- practicality would force us into using some imprecise concept instead, thus opening the door to the sorites.

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