Default filters in records doesn't filter date field in the grid

No, the Search is a text only "contains" search. That is: it will look across each row and where there is a Text value it will look for a match of you search term anywhere within the text values. So you have two choices:

• create a text-based version of the Date in the database (you could create a VIEW on the current table and make the date into a text-representation so that it comes over to the record already in text form). I've found I often end up using VIEWs for my Record List Views as they often need to contain data from across multiple related tables and can allow you to conduct this kind of transformation as well
• you can add a date filter to your record (which is a type of search in its own right)

You can search on date field if it is coming from database. You can't search on  manipulated data in list View. So if date is directly coming from database than you type the date in correct format in search field. The date field can't be pre populated in search field you should have to type manually.

You can create an expression-backed record and make the search box do whatever you need it to do.