Unfortunately, the balance of evidence leans more towards your friend's side of the argument than yours. As plate armor became cheaper to produce, more and more soldiers were outfitted with it, resulting in "munitions plate," armor designed to be mass-produced and issued in large numbers to regular troops. Munitions plate wouldn't be the full-body protective gear that the heavy cavalry might wear (it often consisted of a breastplate, helmet, and protection for the thighs and arms), but it was substantially better defensive equipment than regular footsoldiers of the 13th century would have had access to. This increase in protective gear went alongside with the gradual development of standing armies who needed to be issued with gear (as opposed to previous systems of private retinues and other contracted soldiers acquiring their own equipment). So it's clear that generals did actively attempt to protect their troops better when it became economically possible to do so.

You did make a good point, however. There are in fact battles where the weight of armor became a problem for men-at-arms equipped with plate harness. The Battle of Agincourt, famously fought in muddy terrain, saw attacking French knights become tired and worn out as they slogged across the field to meet the English (who, as the defenders, had the privilege of standing in place and conserving their energy). Lightly armored English archers were able to move more easily in the muck and helped outflank the French after they made contact with the English men-at-arms. At Fornovo, the French royal infantry (including Swiss mercenaries) got the advantage over attacking Italian troops due partially to the difficult conditions of the battlefield. The Italians attempted to attack the French on the march when they were moving across a river. This plan turned to disaster when the Italian men-at-arms found the muddy, rocky riverbanks a difficult place to fight in when wearing full armor. The French infantry were equipped with lighter gear (presumably just breastplates or perhaps brigandines) and successfully fought their way through the Italians to reach safety. Sometimes having a force of soldiers equipped in lighter armor turned out to be an advantage, but that wasn't a deliberate choice on the part of the commanders involved. The French did not issue brigandines to soldiers for the purpose of fighting in difficult terrain; the soldiers wearing brigandines simply found themselves in a position where their lighter gear turned out to be an advantage.