How to change media image private to media image public, just for a specific filemime, and update the url?

In general, you should avoid updating tables directly in favor of using Drupal's APIs, so I would not recommend using an SQL query like that.

I had to do the exact opposite of this recently - move files from public to private. The process is the same, here's what I did:

1. Update the file field's settings to point to the new destination, e.g. from private to public. This will ensure any future files go to the new spot. (You noted that you did this.)

2. Next, we need to update all existing files. To do this, I wrote a post_update hook to move the files for all of the file entities. (This will be in lieu of the SQL query you're trying to use and you also won't need to manipulate the filesystem directly, i.e. no copying files to different directories, etc.) Note that I did use a batch process, but here's the heart of the code:

$new_file_dir = 'public://my-new-file-directory/';

$mimes = [
  'image/jpeg',
  // etc...
];

// Get the file entity ids for the files with your wanted file mimes.
$file_entity_ids = \Drupal::entityTypeManager()
  ->getStorage('file')
  ->getQuery()
  ->condition('filemime', $mimes, 'IN')
  ->execute();

// Load the file entities.
$file_entities = \Drupal::entityTypeManager()
  ->getStorage('file')
  ->loadMultiple($file_entity_ids);

// Move the files associated with the file entities.
foreach ($file_entities as $file) {
  \Drupal::service('file.repository')
    ->move($file, $new_file_dir . $file->getFilename());
}