To me, the core specification is pretty clear that this is required.
From http://hl7.org/fhir/http.html#search :

Because of the way that some user agents and proxies treat GET and POST requests, in addition to the get based search method above, servers that support search SHALL also support a POST based search

However, the IG Publisher tooling that generates Swagger/OpenAPI definitions from a given implementation guide does not include these special POST endpoints.

Further, when I asked about it at https://chat.fhir.org/#narrow/stream/179166-implementers/topic/Search.20via.20POST , one of the US Core authors responded to indicate that Search via POST is not required by US Core and so I think we need some clarification.

Can IGs make these decisions for themselves or must all FHIR servers that support search also support search via POST?

FWIW, my recommendation is this:

• all servers supporting search SHALL support search via both GET and POST as already indicated in the base spec
• the OpenAPI generator included with the IG Publisher tooling should be updated to include the `_search` endpoints for each searchable resource type in each IG