Long story short, even though I've got all the materials and the steel smithing skill, steel plate armor doesn't show up when I use a forge. Do I have to be a certain smithing level, or use a certain forge, or what?

I believe you need the advance armors perk so you can craft Scale and steel plate, both of which will be under the steel category.

Oh, so you do. Why would they make you go down the light armor path to make heavy armor, I wonder?

They probably didn't intend for it to be like that, it probably just got in by mistake. I'm hoping for a fix come thanksgiving.

Steel plate is better than Dwemer armour, and is craftable after you get the Advanced Armour perk, not the Steel armour perk. It is found on the left (light armour) side of the perk tree, after Elven.