To begin with, maces (or hardwood clubs with spikes) were known to the Romans, Constantine had his lighter cavalry use them against Roman Cataphracts at one point and they were known in the East as the best means to defeat heavy cavalry.
You'll note I specifically said "up to 1200" which is before the advent of true "plate" armour. The Western "Coat of Plates" was roughly equivilent to the multi-layered armour worn by Roman Cataphracts that was primarily scale-over-mail. That's a very effective defence, it spreads the impact well.
As to the Roman cavalry vs the "medieval knight trained from childhood" it's important to note that Roman cavalry were also trained from childhood, from peoples with a strong tradition of mounted warefare. These men learned to ride without stirrups and they controlled their horses exclusively with their legs, as both hands were taken up with the lance.
There's a long-standing myth perpetuated by a few academics with much education and no practical experience that the stirrup is necessary to couch the lance, it isn't. The essential piece of equipment is the saddle, stirrups provide lateral stability and they transfer some of the weight of the rider away from the horse's spine, but they do not prevent the horseman from being unseated during a charge, the high pommel and cantel of the saddle prevent this.
What Stirrups also do is make riding much easier, which allows you to start riding later and still become highly proficient, which means you can devote less time to it. In the Roman period cavalry were drawn from cultures with a tradition of horsemanship but cavalry were not a social elite - or rather socially elite cavalry did not form elite units. In the medieval period the social elite.
Medieval knights formed a relatively large social elite and they could learn to ride and fight from horseback relatively late - they weren't as skilled because they didn't need to be. Now, that doesn't mean they weren't still more skilled than most horsemen alive today.
As noted above - the Romans were familiar with shock cavalry, and whilst the couched lance appears to have been unknown I've seen it done by a man riging in a Gallic saddle without stirrups. It's entirely possible that it was known but not used, or that it simply wasn't invented.
Roman-era Contarii and Cataphractarii used lances as long as medieval ones. We don't know if they charged as the gallop or the canter, but we should not assume they were incapable of it. What we do know is that heavily armoured men sat horses as large as medieval chargers and used a 4 metre lance in two hands at high speed, and they did this without the stirrup that we today tend to rely on. I've ridden without stirrups and it's decidedly unnerving not to have your feet secured.
Roman infantry had a number of means to deal with cavalry - in one demonstration I saw them form three ranks. The first rank grounded their shields and plant their pila, the second rank raise their shields over the heads of the first rank and crouch over them. the third rank throw their pila into the faces of the oncoming cavalry.
It's rather like a Napoleonic Square formation. You don't need hugely long spears to fend off cavalry, a wall of shields will generally stop horses, and sharp points will guarantee it. Roman auxilaries carries spears over six feet long, which is about the same as a medieval spear or halbard. Note that the Norman Knights did NOT break the Saxon shieldwall as Hastings, part of the Saxon army charged down the slope and exposed itself to a Norman charge, then they were whittled down by bolts and arrows. It was only at the end that the cavalry broke them.
I would bet Roman infantry against medieval knights every times, and I'd half those odds id you add in the Roman field artillery which would flay they before they even charged home.