$facet current does not report its plan summary stats currently. The problem is similar to the work been done on $lookup, $sort, and $unionWith.
The problem of not reporting such stats becomes more visible when $facet is present as a sub-pipeline and the top-level pipeline is been asked to report its aggregated summary stats. In this current scenario, $facet will not be able to report such stats, and hence, the top-level pipeline will also report incorrect stats.
A client could also ask for stats directly from $facet if it is present as a top-level pipeline and this will also report incorrect stats.
The infra for the same is already available in the current code-base where each derived document source should override the DocumentSource::getSpecificStats() method, if required. SpecificStats base class has a helper method accumulate() which is a convenient method to gather the summary stats from foreign DocumentSources.
In this ticket, we should utilize the current infra and $facet should start reporting its stats.