How do I display the date but not the author? I want articles to show Submitted on without showing Submitted by. The checkbox on Display settings toggles both, not only one of them.
Control/modify display settings (author and date) on posts explains how to do it in Drupal 7.

How should I achieve it in Drupal 8, for example with the Bartik theme?
It's enough to edit the node.html.twig template used from the theme. For example, the Bartik theme uses a node.html.twig template containing the following code. (I omitted the documentation comment at the beginning of the file.)

{%
  set classes = [
    'node',
    'node--type-' ~ node.bundle|clean_class,
    node.isPromoted() ? 'node--promoted',
    node.isSticky() ? 'node--sticky',
    not node.isPublished() ? 'node--unpublished',
    view_mode ? 'node--view-mode-' ~ view_mode|clean_class,
    'clearfix',
  ]
%}
{{ attach_library('bartik/classy.node') }}
&lt;article{{ attributes.addClass(classes) }}&gt;
  &lt;header&gt;
    {{ title_prefix }}
      {% if label and not page %}
      &lt;h2{{ title_attributes.addClass('node__title') }}&gt;
        &lt;a href="{{ url }}" rel="bookmark"&gt;{{ label }}&lt;/a&gt;
      &lt;/h2&gt;
    {% endif %}
    {{ title_suffix }}
    {% if display_submitted %}
      &lt;div class="node__meta"&gt;
        {{ author_picture }}
        &lt;span{{ author_attributes }}&gt;
          {% trans %}Submitted by {{ author_name }} on {{ date }}{% endtrans %}
        &lt;/span&gt;
        {{ metadata }}
      &lt;/div&gt;
    {% endif %}
  &lt;/header&gt;
  &lt;div{{ content_attributes.addClass('node__content', 'clearfix') }}&gt;
    {{ content }}
  &lt;/div&gt;
&lt;/article&gt;


The line printing the name of the user who created the node and when it was posted is the following one.

{% trans %}Submitted by {{ author_name }} on {{ date }}{% endtrans %}


It's enough to change it and remove the part printing the username. In the case of the Bartik theme, the node.html.twig content will become the following one.

{%
  set classes = [
    'node',
    'node--type-' ~ node.bundle|clean_class,
    node.isPromoted() ? 'node--promoted',
    node.isSticky() ? 'node--sticky',
    not node.isPublished() ? 'node--unpublished',
    view_mode ? 'node--view-mode-' ~ view_mode|clean_class,
    'clearfix',
  ]
%}
{{ attach_library('bartik/classy.node') }}
&lt;article{{ attributes.addClass(classes) }}&gt;
  &lt;header&gt;
    {{ title_prefix }}
      {% if label and not page %}
      &lt;h2{{ title_attributes.addClass('node__title') }}&gt;
        &lt;a href="{{ url }}" rel="bookmark"&gt;{{ label }}&lt;/a&gt;
      &lt;/h2&gt;
    {% endif %}
    {{ title_suffix }}
    {% if display_submitted %}
      &lt;div class="node__meta"&gt;
        {{ author_picture }}
        &lt;span{{ author_attributes }}&gt;
          {% trans %}Submitted on {{ date }}{% endtrans %}
        &lt;/span&gt;
        {{ metadata }}
      &lt;/div&gt;
    {% endif %}
  &lt;/header&gt;
  &lt;div{{ content_attributes.addClass('node__content', 'clearfix') }}&gt;
    {{ content }}
  &lt;/div&gt;
&lt;/article&gt;


For a different theme, it's enough searching the line of the node.html.twig template containing {{ author_name }} and remove it. Clearly, also the sentence containing {{ author_name }} needs to be changed to make it grammatical. (Submitted by on {{ date }} would not be grammatically correct.)

As general suggestion, never change a template file used from a Drupal core theme or a third-party theme. Rather, create a sub-theme using that theme as parent theme, and add the template files you need to change.