Probably not. The problem with current search engines is that they need some way to rank the pages and hence inherently susceptible to clever SEO (that curates pages to get ranked high for certain type of queries).
So having a paid search engine does not fully solve the SEO problem. Having no Ads does not takeout the SEO problem of boosting pages to top.
If the paid for search engine ignores SEO "optimizations" and actually ranks on content AND these results prove better that attracts actual paying users, then we don't have to wait for SEO to die. Just like SEO rose to prominence to win Google result ranking because that's where the users were, the sites would stop the SEO crap.
Getting this hypothetical site to get users is the real problem. Same thing with getting users to a Facebook alternative.
So having a paid search engine does not fully solve the SEO problem. Having no Ads does not takeout the SEO problem of boosting pages to top.