Plumbing – How to snake this tub drain

First, make sure that the clog is accessible via the overflow. Try this test... While the tub is draining slowly, pour water down the overflow through a funnel. If that drains fast, then the clog is in the horizontal arm and no clearing through the overflow will help you.

What works for me is to use a wet cloth to seal off the overflow (replace it first to make as small an opening as possible). With the tub 1/2 full of water and holding the wet cloth over the overflow vent, a plunger should then be able to clear the clog because it will force water down the drain (horizontal or vertical) rather than pushing water up the overflow pipe (or pulling air down it). Should that fail, use draino or physically pry off the grate for an auger.

The plunger should pull some amount of gunk back up into the tub while you're using it, even through the grate. If that doesn't happen and the drainage does not improve, then you may have a clogged vent. You need air behind water for it to drain quickly otherwise it's like water draining from an upside-down bottle. Cleaning the vent means going up on the roof and clearing the clog from there -- a hose at full blast will usually do this.

As for fully removing the overflow and parts... I don't know.

I would stay away from the acid because it can also eat your metal pipes. If the drain hasn't been used for years it is possible that it's rusted in place. I would try CLR as mentioned above. The balloon trick is clever. Slow process but it MAY work. Most older drains are brass and the stopper is also brass. Corrosion is more likely the culprit. Once you get it free you can pull the entire assembly out through the overflow hole to clean or replace it. Good luck!