This functionality is already implemented in the operating system's test version.
Microsoft is working on reintroducing the search bar to the Windows 11 taskbar. This feature was not included in the Windows 11 release version, but recently appeared in the Windows 11 preview build.
According to Tom's Hardware's screenshots, the search results will be displayed in a pop-up window. You may, for example, view only apps. The window appears to be the same as in the current version of the OS.
OS users will be able to restore the Taskbar's previous behavior if desired. An option will appear in the Taskbar's personalization settings that allows you to substitute the search box with a button in the spirit of the current OS version.
Microsoft has not yet revealed when Windows 11 will get a new Taskbar.
This is an interesting piece of information.