Hi Janos Torba,
That is, because that input is not the right one. That one is used for visual purposes only. The actual value is taken from the hidden input just below the visual input: https://i.imgur.com/H2PdowG.png
Something like this will do it:
jQuery('input[name=post_date_from]').val('2022-03-01')
The search will always take the parameters into account, the form values from the settings is always sent as a serialized array to the back-end.