Coat hanger?

A problem with the info you gave.

You have the tailpiece out but say the hole is rusted solid; that's hard to do in porcelain. The tailpiece is also most likely brass, both the sleeve and the big nut underneath. No rust there either. Is all of the tailpiece out?

Remember that replacement tailpieces come in varieties that may or may not have a hole where that hole in the porcelain is, and if you do use one of those "solid" wall tubes, the overflow will not work and may clog hard over the years..

Thanks. It's too solid for a coat hanger so I think the overflow feature is permanently dead. I can stick something through the upper hole pretty far; it seems to be jammed at the opening only.

Not sure, Grubby. I'm sure the old drain assembly is out. The old tailpiece has a hole although it might not have been lined up correctly. Maybe it isn't rust. It's black and hard and smelly which isn't exactly a characteristic of rust, but there were rust stains around the drain so it's rusting somewhere. (The sink is cast iron.)

I don't feel the need for an overflow, so I don't NEED to save this function. Is it even possible to stop up the top overflow so that no more water gets in?

I would go to a home improvement store like Lowes and ask the clerk in that department. They are usually very knowledgeable. I googled, but only found solutions for bathtub over flow. It suggested a snake for that job.

I just had to clean this out on my sink (installed 2008) I used a piece of spring brass wire, about 0.96mm IIRC. It is harpsichord stringing wire. I didn't want to use steel piano wire because it could scratch. I fed it down through the three overflow holes. It had a small loop bent on one end , and then I used an old toothbrush from the drain side once the plug of disgusting slime was broken.
I read online that the black slime grows because of Listerine. Go figure. I put some lysol TBC down the overflows, hopefully that took care of the rest.
Casey

Thanks for the suggestions. I can poke at both sides of the clog and it feels like concrete. The drain hole area in the sink is really corroded at the level of the overflow outlet which makes me think that even if the overflow could be unclogged, I should not be introducing water to that area anyway. Looks kinda like this (the black spots in the enamel are crud, not missing enamel, but inside it's that bad:

and the old piece looked like....

So I should not be using the new tailpiece with the overflow openings I think...any suggestions on keeping this thing going a few more years?

In conclusion, I used a chisel and hammer and knocked out all the crud. The enamel actually looked pretty good under it, nice and pink. It almost seems like someone had intentionally filled the entire overflow section of the sink throat. Maybe it's 20+ year old plumber's putty. It was icky.

Weird.

Well, that was fun. :)