Pools search is restricted to Name, Descriptor, Creator and Status, Category and ordering options.
This makes it far less powerful than post searching.
On the other hand, finding pools in posts means seeing every post in that pool, individually and not being able to filter by any of the methods afforded by the Pools tab.

I think these 2 features need to be combined somehow.
The only quick implementation I can imagine is having pools tagged with every tag found within the posts of that pool, so that pools can be searched using tags.
Otherwise there needs to be some filter you can apply which only shows the first post of a pool that fits your criteria, so you only get 1 entry returned per pool in your search results.

Those 2 changes would bring the power of post search tags to Pools, as well as the condensed view of Pools to the post search, as imperfect as it may be.

An addition of a User tag for Creator, a Text Searching tag for Pool Description or Name, a PoolActive tag for Status, and a Series/Collection Category tag would also be helpful.