[Agema]

If knights still had a useful function on a battlefield in a professional army era we should expect to have seen large bands of mercenaries taking over that role.

I think we're maybe confusing two issues: the social role of a feudal knight; and the battlefield function of a knight, which is to say heavily-armoured cavalry.

The rise of mercenaries and professional armies was largely about the social limitations (and therefore eventual decline) of the feudal/chivalric system. That's mostly an off-battlefield issue, and hence the decline of the *knight*. On the battlefield itself and occuring roughly coincidentally were the technological advances that made *heavily-armoured cavalrymen* obsolete.