Drupal 8: Node form submit/update - how to redirect to a custom page with an code-based approach different than hook_form_alter()
While the other solution is perfectly valid, I mark this answer as accepted since it comes closer to what I was looking in first place - the new way to alter forms.

Eventually the solution can be found right here:
https://www.foreach.be/blog/how-manipulate-forms-drupal-8 

In summary (quoted from the article):
1 ) The first step is to write a service that intercepts the original UserLoginForm and directs to the adapted form

2)  Changed the route to return the adapted NewUserLoginForm instead of the original UserLoginForm

3) The new form extends the old form and inside the buildForm method you add in the beginning 

$form = parent::buildForm($form, $form_state);


This step was missing in the example tere