Flag count is incrementing only after page reload
I had the same problem, and find how to solve it. The Views Flag Refresh module is only available for Drupal 7, but someone ported it to work on Drupal 8, here is the download link.
I find it in this issue comment by WalkingDexter.
In my case I need it to work on Drupal 9, so it is necessary to change a line in a file in the downloaded custom module as Akash_Daimari explains in this comment.
The module would go in /modules/custom/
In the view you have to use Ajax and add the flag in 'Flag refresh' options.
As a note, in the View preview the count does not update, but on the page it does.