This is more an argument that mechs will never be dominant on the battlefield, not that they won't exist at all.
One of the painfully learned lessons of modern war is the necessity of "combined arms," of more than one type of warrior, working together. Even the most powerful, heavily armored tank has vulnerabilities -- e.g. poor vision and lack of fine-grained situational awareness -- that the humble foot soldier can cover. When you fight a combined arms team, one of the things you really really want to do is to separate their armor from their infantry, because defeating them separately is easier than defeating them together.
As long as there is some necessary task on the battlefield, for which mechs are better adapted, mechs will have a place. If history is a guide, there will be such a role.
Even in ancient times, armies combined infantry, cavalry, archers, etc. I wouldn't jump to the conclusion that it will be different in the future.