Yes, every self-biased circuit needs a startup, and that covers pretty much all bias circuits that I know of.
You can do a DC sweep of the current in the 2 branches of your bias circuit to see where the circuit has stable
DC operating points. You will see that your circuit probably has a second operating point for the case where
no current flows into either branch.
This is actually a metastable state but even then, the circuit is unlikely to startup well without a
well-designed startup circuit. It is worth the effort to make this, and to make it well unless
you want to make it after your silicon is back
Regards
Vivek