Why not shutdown Yahoo all together?

Because Verizon Media chooses not to shut it down. It may be that they are receiving enough revenue from other areas of Yahoo to keep it worth their while to maintain (for now). It may be that shutting down parts at a time benefits them better than closing the entire brand/property. Only corporate knows the reason(s) for their choices.

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YA is most likely being targeted because of the REALLY low quality of content in recent years and the amount of money it would cost to clean up the site and get it running well enough to try to attract more quality users AND advertisers willing to spend decent amounts of money to advertise on it. Since there are a number of sites that are competition of a sort (many more than there were when YA was originally created) it's likely just not what Verizon wants to do with their money, and they don't expect improving YA would generate enough of a return to make it worthwhile.

I never use Yahoo's search engine.

I don't know. Nobody uses the search engine, especially since it's a shitty Bing clone.

People should use Google or DuckDuckGo instead.