Do you have a packaged workbook with sharable data that can be shared to show how the search was implemented?
If your data is sensitive, can you mock up the pagination and search functions with sample data?
We had raised a case with tableau support regarding this issue .
we have got the following reply from them that pagination is not native to tableau and what we had achieved is the closest that we can get . and W.R.T the slider not refreshing to page one this is what the engineering team had to say.
update from our Engineering team that the navigation button behavior on the slider filter is expected behavior due to the results of the action filter on the table calculation. In other words, the navigation button gets disabled because Tableau Desktop is told there are no other values to navigate to, but the slider bar will allow going back to page 1.
As an alternative, using the Single Value Drop Down list option is cosmetically similar and allows the user to navigate back to page 1 from page 2.
Hence marking this post as answered tableau should provide for a native component for pagination.