Generally the drains that are controlled from the
OVER-FLOW do indeed have a rod that goes down
from the face of the overflow... which lifts (opens)
the drain by lifting a hollow (typically plastic) sleeve
from in front of the outflow of the drain...
It is hollow to allow the OVERFLOW to work at all times.
The version you seem to have... WOULD have a
bellows like device that pushes on a plug lever...
More complex... but I seriously believe both may be
serviced through the OVERFLOW
(will attach a drawing below)...
you will want to block the drain before rolling & dropping screws in, on & around the drain HOLE
Once you see the diagram of the drain MECHANISM...
you will be able to see that jamming
in a coat-hanger
was a terrible idea.
You can still most likely remove and repair the
mechanism ... (from the over-flow)
You are looking at buying a new one and installing the entire drain from the bottom (hopefully).
Diagram of drains:
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Carnac the Magnificent