No, Youtube should not own Google, because Google has run Youtube poorly since taking over. Youtube has become all about profit. This is never more evident than when Youtube allows sham companies to make false copyright claims against individual owners. This is all a scam for profits, and Google is about profit, so it does not care.
Youtube is popular for good reason, however Google has an omnipresence that is almost unbelievable. From Operating systems for cell phones to search engines and marketing, Google is making smart investments for the future. If Youtube had done the same, then maybe it would have owned Google, instead of the other way around.