What am I doing wrong in my theme that's breaking CAPTCHA?
I found the conflicting template inside USWDS base theme. For anyone else having this problem, you must alter this template:
/themes/contrib/uswds_base/templates/form/details.html.twig

You can either wholesale replace with the same template from Bartik's classy template (or similar), or else custom rewrite it using the working template from Bartik and customizing it to what you need. I needed to rip out the ul/li based markup that USWDS imposed, but still keep the classes and accessibility markup.
In case it helps anyone else, here's what I ended up using:
&lt;div{{ attributes.addClass('usa-accordion usa-accordion--bordered') }}&gt;
  {%- if title -%}
    {%
      set summary_classes = [
        required ? 'js-form-required',
        required ? 'form-required',
      'usa-accordion__button',
      ]
    %}

    &lt;button type=&quot;button&quot; aria-expanded=&quot;true&quot; aria-controls=&quot;{{ attributes.id }}-content&quot;{{ summary_attributes.addClass(summary_classes) }}&gt;{{ title }}&lt;/button&gt;
  {%- endif -%}
  &lt;div id=&quot;{{ attributes.id }}-content&quot; class=&quot;usa-accordion__content usa-prose&quot;
&gt;
    {% if errors %}
      &lt;div class=&quot;form-item--error-message&quot;&gt;
        &lt;strong&gt;{{ errors }}&lt;/strong&gt;
      &lt;/div&gt;
    {% endif %}
    {%- if description -%}
      &lt;div class=&quot;details-description&quot;&gt;{{ description }}&lt;/div&gt;
    {%- endif -%}
    {%- if children -%}
      {{ children }}
    {%- endif -%}
    {%- if value -%}
      {{ value }}
    {%- endif -%}
  &lt;/div&gt;
&lt;/div&gt;