What would be the output voltage of the full wave rectifier output when center-tap is removed?
The centre tap is a low impedance source coupled to the primary low impedance.

The load R's are fairly high impedance.  When the caps  and load R's  equal, there is no difference.

To keep the voltages balanced with a mismatched load, the source impedance must be low.

We call this mismatch from loads, load regulation error where the error is due to the voltage divider % (Rs/(Rs+Rl)) that is if one knows the source and load equivalent R's for transformer and diodes.