I would like to center to 0V (remove the DC component.)
Centering the signal's amplitude (peaks) around 0V doesn't necessarily remove the DC component. It only does so if the signal's average value - the DC component - is exactly at 50% of the amplitude - i.e. the signal has to be symmetric.
The AC coupling network you devised does in fact remove the DC component - it just doesn't center the amplitude around 0V, because the DC component wasn't 50% of the amplitude.
To get a most generic "amplitude centering" circuit, you need to detect the envelope of the voltage, and then shift the signal accordingly so that the center of the envelope is at 0V.
For example:

simulate this circuit – Schematic created using CircuitLab
The output signal at B is indeed centered around 0V:

A more practical circuit of this kind would use rail-to-rail I/O op-amps, perhaps a bit faster than TL084 to improve accuracy.