Once we go back to period of 500 BC and 1000 AD, a notable period of time, the single most prominent melee weapon of a cavalryman has been the sword. Greek companion cavalry, west roman equites, east roman comes, anglic war chariots, alani horsemen, persian camel riders all of these favoured one handed melee weapons, swords or axes mostly, precisely because spears and lances could break and leave the warrior with... you guessed it, with only their secondary weapon, a sword or an axe.
Out of all polearms, spears and lancae were the uncontested favourites.
Pollaxes didn't come into picture before the turn of the 14th to 15th century and by then it was an infantry weapon wielded by armored combatants to defeat other armored combatants.
Look.
I understand that the sword was probably really common amongst cavalrymen. Infact... I know it was. But it simply wasn't the most common primary weapon used by ancient cavalry. It might have been, if you're specifically JUST talking western cultures (Roman, Greek, Persian, Briton, etc) or even more modern cavalry (Civil war cav typically used pistols and a cutless) but it absolutely
was not the most common
PRIMARY weapon cavalry used throughout all of human history. In fact - the flail was more common than a lance - if we're talking the crusades. The crossbow was the most common cavalry weapon - if we're talking ancient china.
But the simple fact is the most common weapons used by ancient peoples using horses as transportation is 1) 2-handed polearms. (Lances, Spears, Haldberds, Poleaxes). 2) Bows
You said it yourself - the reason swords were popular was because spears and lances broke.
That would mean the sword is a BACKUP. They'd prefer another ****ing lance. It's safer. Your enemy is further away. It's the same with bows - If I'm riding a chariot I'm going to shoot every damned arrow I brought before switching to my polearm - and I'm going to use the **** out of that polearm before I switch to a sword. I'm on a damned chariot - they can't hit me with their swords unless I'm trying to hit them with mine.
Also, the primary weapon of Alexander's companion cavalry was the lance. Specifically the xyston, a double ended cornel-wood lance.
Roman equites used an extra-long sword (machaira) or broad sword (spatha) but also relied heavily on javelins (so much so that they brought three quivers of them - and used them as lances when need be)
The Roman Cataphracts primary weapon was the lance.
Starting from the EARLIEST Achaean Chariots introduced to Greece - The spear was the primary weapon.
Alani horsemen carried shield and spear into battle.
Persian Dromedarii used the spear as a primary weapon.
I'd list my sources, but I really don't care enough to put any more time or energy into this. Just know that saying "Swords were the most common weapon of cavalry" is wrong. It isn't, no matter how much you want it to be. It doesn't make sense.