As you can see, I've researched siege equipment. For some reason, they never register in sieges. 

I have no clue what could have caused this. 

Someone suggested on Reddit that siege engines reset each era. For example, OP's screenshot shows that they are in the Industrial Era, so they can only build the siege engine for that era. This is a poor choice in my view; it's not as if later cultures forgot how to build trebuchets, and you can continue to build ordinary (permanent) units until you research a tech with a superior unit in that class.

The extra twist here is that I don't think there is a siege engine for the Industrial era: confusingly, the Siege Artillery unit is  a Heavy Weapon class unit, not a Siege Weapon class unit.

So in the Industrial Era, you are supposed to bring pre-built Siege Artillery weapons from your homeland (once you have the right tech), rather than building Siege Weapon-class units after the siege has started.

(It seems unlikely that OP will be doing this any time soon since it's the 7th century AD and their army is mostly using crossbows, but the fact that you can reach the Industrial Era in the first millennium AD is obviously a separate bug.)

All this makes historical sense but if it's not documented anywhere.

The root cause of this bug report is the Devs' decision not to write the in-game encyclopedia until just before release.They should have written an in-game encyclopedia that picked up data directly from the game files, so that beta testers would have the facts to hand even if they changed rapidly. Then these forums would contain more useful feedback and fewer puzzled players. ?