I think every classes need 3 skills that have little or no requirement, and 3 skills that require something. For example, i want to play cataphract but dont want to use spear. there are 2 skills that can be used without spear but remaining 5 skills points become pointless 
I think what you want is just cataphract skills being useable with more type of weapons.
It’s called choice of life.
You will need to stick to a weapon type or two max.
Catphract is a LANCER build. Aka he uses spears ^^.
Get another class if you don’t want to use spears~
There’s lots without restrictions.
And your unable to use most skills while being mounted (without auto unmounting~).
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a lot of classes kind of do. at least I know quite a few class lines that do hoplite 1 with stabbing/pierce, highlander with everything but the blocking move in C1 anyway, all swordsman skills, a lot of the barbarian skills just to name a few the other I think are for obvious reasons shield moves for example and the cataphract skills are no exception.
Impale would require a spear to use as you would not have the power to impale someone like that one a sword. maybe a 2 handed sword but defently not a normal sword and swords are more for slashing than stabbing unless we are talking about rapiers and even then that would not be able to impale a target like that.
Earthwave ok I feel earth wave could look useable with a 2 handed weapon but I would not say a 1 hand sword or rapier exception to 1 hand spear as it got the reach to hit the floor
Trot and Steed charge have no requirements.
Doom Spike. could be more compatible I guess. mostly for thoughts of a rapier as its a stabbing move which would not really require as much reach as other moves.
Rush. This one would make no sense with any other weapon but spears. its a lot easier to spin a spear like that than other 2 handed weapons. and I could not see a 1 handed weapon again exception 1 hand spears having such reach to make it useable. and a 2 handed sword… well your more likely to chop your arm off than the enemies lol.
in conclusion I think the reason a lot of the moves require a spear is based on the height elevation of the mount or the type of actions your doing on a mount . remember the mount is about half your size. then your sitting on top of it. this is why both here and historically spears were the favoured choice of weapon with a mount. they had reach. and the way spears are designed worked really well for rush attacks . they could make the cataphract more appealing for other weapon types but I feel that would require them having to make more skills to make it appear for those weapons as with cata’s current weapon skill sets it makes sense