Bodkins are used against maille, not plate; the needle like point would slip between the rings and stick in the target. Against plate armor, the arrowhead was more short, heavy, and was wider than the shaft of the arrow, so the whole thing could keep going without losing momentum to friction. Mongol arrows were typically shot at unarmored game during hunts or quite lightly armored men during war, so armor piercing arrows weren't a priority.

As to whether their bows could pierce plate armor, it isn't really known, since plate armor wasn't in full swing during their forays into eastern Europe, but good plate was usually sufficient to protect the man wearing it from English longbows, which were similarly weighty to composite bows used by the Mongols.

I have seen examples in museums of Mongolian arrows with extremely thin points, which exist across the world in order to defeat mail armor. They could be considered in the bodkin class, but I'd be surprised if the technology moved from Western Europe to Asia, and wasn't just a common innovation against armor like male which was common across the world.

There are a few other points that should be brought up, the height of the Mongol conquests happened before plate armor came into prominence, with Genghis Khan dying in the early thirteenth century. The Mongols would have been facing mail with some reinforcement like leather or small steel plates, but they wouldn't have been facing full Italian plate armor. I would not agree completely with dandan noodles when he says that the Mongols didn't prioritize armor piercing arrows, as the Mongols fought a huge array of different opponents, and by this time armor was very common. Like all successful militaries at this time, they would have had different arrowheads for fighting different troops.

The other point to consider is that bows are relatively simple instruments. They transfer the energy from the draw into the arrow and propel it forward. While both the longbow and the Mongol bow have expensive mythology surrounding them, the simple fact is a bow of a certain draw weight will perform very closely to another bow of a similar draw weight. So a Mongol bow of 100 pound draw weight would function essentially the same as a longbow of 100 pound draw weight.

Finally, there's the point that bows were generally not well equipped to penetrate solid steel plates. There are some demonstrations you can find on youtube, or in the weapons that made Britain series that demonstrate some penetration ability. However, none of these penetrations are fatal, and are almost always in ideal circumstances. Simply put, no bow would be successful in lethally penetrating steel plate, at least not reliably.

Bodkin arrows always get mentioned whenever anyone praises English archers but it's always overdone. The English didn't invent them. What do you think the French/Genoan crossbowmen were shooting back at the English? The points on their bolts look just like English bodkins/lozenges. 'Bodkins' could be found in just about every archer's arsenal the world over.

Small, thin arrow points are obvious, even before armour penetration comes into consideration, small points are more convenient and more accurate. Broadheads do more damage but they're not without their drawbacks. They're heavy and awkward, somewhat difficult to have ready in quivers. So even if you weren't used to attacking heavily armoured opponents, you would find narrow points useful.