First the supply issue : The biggest reason for Horse archer scarcity is the training curve in becoming a mounted archer. It's not a simple task, you need to be able to ride and control your horse with only your legs while drawing a bow and accurately firing all while the horse moves...not a simple feat by any means.
Roman (and Greek) society were founded in cities...infact most 'empires' excluding the mongols were not nomads and had settled into cities. Although they made use of horses, much of the population were not active riders. When you get into medieval times, only the wealthy had the access to horses and time to train.
Mongol society is heavily different than other empires...it's people were heavily nomadic and as such, much of the population were actively riding horses on a daily basis. There is the saying that a mongol warrior was born on horseback...not quite accurate, but many were already proficient equestrians by the ages of 3 to 4. This gives the mongols an unprecedented population of riders to recruit from that already possessed horse archery skills, while other empires would have to actively train these people.
Second is the tactics
Much of Mongol success was also found in surprise. They marched incredible distances in amazingly short time frames and struck enemies before they could organize an effective resistance. However once the defenders started adapting their tactics to be 'anti-horse archer', the mongol forces began suffering some pretty heavy losses. Although the horse archers were dominant on the open field and in raid settings, they don't really add much to a siege and a fortified town often proved resistant.
So, if Genghis Khan's military was so superior to European knights
I need to challenge that line as the answer here is...they weren't. They had an extremely easy time dispatching Hungarian and Polish defenses in their first round of invasions/raids, however Hungarian and Polish forces were mainly light cavalry and not what we would call 'knights'. Knight orders actually fared decently vs Mongol horse archers. The first defeats to the Mongols taught Polish and Hungarian leaders a couple lessons, in particular that "Two elements of the Hungarian defense had proved effective, however: close combat with mass armored knights and stone fortifications"."
https://en.wikipedia.org/wiki/Second_Mongol_invasion_of_Hungary
King Béla IV took note of this, and spent the next few decades reforming Hungary in preparation for possible future invasions. He used a variety of methods to do this. First and foremost, he amalgamated the servientes and iobagiones castri into a new class of heavily armored, well-trained knights of the western type, where previously Hungary's defenses had relied almost entirely on wooden castles and light cavalry.[10] In 1247 he concluded a feudal agreement with the Knights of St. John, giving them the southeastern borderland in exchange for their help in creating more armored cavalry and fortifications.[citation needed] In 1248, he declared the country's middle strata could enter a baron's service, on the condition that the barons lead the men on his land properly equipped (in armor) into the king's army. Documents from the time state that "the nobles of our country can enter into military service of bishops in the same way in which they can serve other nobles". After 1250, free owners of small or middle sized estates serving directly under the king were included (along with barons) in the nobility. Finally, new settlers were given "conditional" nobility in exchange for the requirement of fighting mounted and armored at the king's request.[11] In 1259, he requested that the Pope put him into contact with Venice, as he wanted to hire at least 1,000 crossbowmen (crossbows having also proven a very effective weapon against the Mongols, despite the relatively small numbers of them actually deployed by the Hungarians in 1241).[12]
To cement his new defense doctrine, the king offered grants and rewards to cities and nobles in exchange for the building of stone fortifications. The reforms ultimately paid off. By the end of his reign, Béla IV had overseen the building of nearly 100 new fortresses.[13] Of these 100, 66 stone castles built on elevated sites.[14] This was a major upgrade from 1241, when the kingdom only possessed 10 stone castles, half of which were placed along the border with the Duchy of Austria.[15]
In short...when you are facing wooden walls and lightly armored horsemen that aren't aware you are coming, horse archers do amazingly well. When you are facing an enemy with stone castles and heavy cavalry, your horse archers have little they can do. After these changes were made, engagements between Mongols and Hungarian soldiers were solid wins for the Hungarians with heavy losses to Mongol forces.
why weren't earlier powers, like Rome, crushed by similar weapons and tactics?
Rome was actually decently resilient to these tactics as anti-personnel siege weaponry was a Roman specialty. Roman Legions would posses a large number of ballista, scorpians, and other large crossbow like weapons that would wreck havoc on horse archers. The specialties and discipline of the Roman legions was somewhat lost in medieval times, what they transitioned to were much more prone to horse archer warfare.
