Why do Op amps need dedicated grounds to source?
I don't understand why the bias currents wouldn't just flow into the
negative terminals of the voltage source even without a dedicated
ground.

They would. They will find a way to flow (out of the bases) into a more negative voltage whether it be ground (0 volts) or the negative rail or somewhere slightly more negative that the common-emitter point of the transistors.
If there is no connection to the bases or, the connection is to a voltage higher than the common emitter node, then bias currents (for this particular op-amp you sketched) won't flow.