Its actually quite easy doable if you are very aggressive. Put star ports at every colony and have a flexible force flying around, make note of where they will attack and use stations to snare them towards the port. I barely had mining and science stations at the start to prevent the funneling. If you do that at the beginning you can purge at least 4 species at a very quick note. Do declare war on them though, do not let them advance beyond cruiser level. Once that is done, its just a matter of snowballing.
Only downside is that the annihilation bombardment is useless and the unity hit.
Here is the screen after destroying the unbidden, purging the fallen empires and destroying the two federated star empires 's army. its just a matter of mopping up now.
Tip: tech requirement and unity becomes massive, so purge helps you with that. For tech, wait for exploring the systems until you have adopted the science tree with the Planetary Survey Corps. That way, the science will snowball immensely at late tech