Get rid of calls to `set_request_cache_with_short_name`

This method from the toggle legacy classes should not actually be
exposed to all. So we get rid of it by manually setting the cached
value. While we are at it, we convert the STORAGE_BACKING_FOR_CACHE
legacy waffle switch to its modern version. As the flag is not being
used elsewhere, this should not break anything.

We take the opportunity to modernize waffle switches from
block_structure.config: to do so we convert the INVALIDATE_CACHE_ON_PUBLISH and
RAISE_ERROR_WHEN_NOT_FOUND waffle switches from legacy classes to their modern
equivalents. These switches are not used outside of edx-platform, so this
change should not trigger any error.