There are about 3 major speech search engines out there. Apple's Siri, Amazon's Alexa and Google. Since you ditched Google and I don't see how to get use of the other 2 on Android unless you move to a Kindle or iPad, it's dead Jim.
Now you may be thinking of something like Dragon Dictate but that doesn't scale down to the smaller tablets so for now, no.
I do encounter folk that "hate" google, microsoft, apple and such and they end up burning up over trying to do this or that without these companies. I guess you could found a new company, create web services and use those.
I no longer have a Google account and have disabled all Google apps except for one called Google App.
I found, after disabling Google App, I have no voice search in the Internet app that came installed on my Galaxy Tab A.
Was looking for a browser that has a voice search independent of google.