Absolutely yes - all hosts, new and existing - should have to either prove ownership, or submit proof of consent from the legal owner, before being permitted to list on Airbnb.
Similarly, In regions with regulatory requirements, all hosts should be required to submit evidence of compliance with local laws before their accounts go live.
Those two simple requirements would immediately eliminate the vast majority of illegal listings on the platform, eradicate the huge problem of anti-social Airbnb guest behaviour and neighbourhood disturbances in over-saturated towns and cities around the world, and return vast swathes of housing stock to local long-term rental markets.
However, rather than acting like any responsible, accountable or ethical company would do by imposing such basic requirements, Airbnb prefers to keep insisting that they bear absolutely no responsibility whatsoever for anything that anyone chooses to list on their platform, and continue to spend countless millions on court battles in the US and the EU, battling for the right not to have to do so.