Organize template logic into multiple Twig files to simplify theme overrides

When I apply this patch on a fresh D11 install via DrupalPod and try to use the filters summary in the Header of the Content view, an Ajax error is thrown and the Header remains unchanged.

If I switch back to version 3.2.0, there is no Ajax error and the summary renders as expected.

@bhogue thanks for the feedback.

I'm a bit surprized about that Ajax error caused by this issue's MR because the generated HTML should be strictly identical.