When you go to the “People” tab in the phone UI and search for someone after you call them/send them a message when you again go to the People tab the search will still be active.
IMHO that search should reset in 99% of the use cases and therefore that should probably be the default behaviour, I do see that there is a use case where maybe you searched a family/company name or similar and want to one by one call all of them but I think for most people most of the time this is not what they are doing.