How to increase the performance, if one of the main activities on a website would be searching in Views exposed filters? Will the Search API module (with Database Search/not Solr) significantly increase the performance of such searches?
It probably will, depending on the Search API backend you are using. Something generally true is if you are using any other backend than database, the querying for the search will be dispatched to this additional backend instead of the database, which is Drupal primary storage, and will avoid your database a lot of suffering.
It will also makes things easier to scale, since a lot of search engines can be replicated more easily than database. Even without this statement, you can at least put the SQL server and the search engine on different servers, which means they will eventually be faster since they won't have to share one box resources.
It doesn't mean it will be faster in all cases, it depends on your environment and configuration.
Regarding the queries themselves, if you are doing a lot of fulltext searches, using a backend such as Apache SolR (or any other search engine, Xapian, etc...) it will surely be a lot faster since they have been made for this. If you are doing facet oriented search, it will also help a lot.
But, in the end, if your views filter only are on exact strings, integers and such, it may not be faster in the end, it will just be more scalable.
You have to remember that creating an index for your data will also imply that you install a whole lot of modules in your Drupal site, you'll have to benchmark a bit. But if the primary business matter of your site is searching, I'd say give it a try.
I don't know if my answer sounds good.