and now you are taking the lasts part of the 14th and the 15th century and using that to talk about a about 500 year period.
Or you don’t know what a man-at-arms is. The term is used about the professional soldier who was trained and equipped to fight mounted during the period from mid 11th century and to the end of the medieval period.
Look at the 1200th century. swords was the main weapon (+ the lance when mounted) and the is still the case 200 years later. That is clear if one study frescoes, and other illustrations.
infantry used spears, peasant levies used spears… because it is cheap and can turn an untrained man into something useful if massed. Usefull on the rare medieval battlefield.
But large battles where rare. By far the most fighting a man-at-arms would do would be raiding, skirmishes and similar, where long pole arms is not that useful, since they require a closed order formation to work.
And the shorter specialized polearms are only relevant when armor get to a point where they are a clear advantage over a sword and shield. … when facing other similar armed opponents.
A polearm (other than a lance) is useless when mounted. And a men-at-arms is trained and equipped to fight mounted.
And we have the everyday wearing of swords.
Then as armor improve we start to see different types of short polearms in use by the professional men-at-arms… when fighting on foot.