how to extend FormBuilder::processForm from multiple API endpoints in one class
A. EntityFormBulder::getForm is just a shortcut for instantiating the form object, injecting the entity and building the render array. You can split this up:

  public function getForm(EntityInterface $entity, $operation = 'default', array $form_state_additions = []) {
    $form_object = $this-&gt;entityTypeManager-&gt;getFormObject($entity-&gt;getEntityTypeId(), $operation);
    $form_object-&gt;setEntity($entity);

    $form_state = (new FormState())-&gt;setFormState($form_state_additions);
    return $this-&gt;formBuilder-&gt;buildForm($form_object, $form_state);
  }


B. You have to inject the entity before you can build an entity form, see the steps above.

C. _entity_form is for integrating an entity form in a route, then the route parameter is upcasted and injected into the form automatically, see How to Inject entity object in EntityForm

The best approach is C. (standard pattern) and refactor the OOP code.

BTW if you don't find a OOP solution or if it is too complicated then you could simply add a custom process callback to $form['#process'] in a form alter hook to all or specific entity forms.