"If some people make political statements, does it keep water from being delivered? Do the relief workers stop?"

Yes.

When politicians get attacked by the media they most often immediately drop whatever they are doing and go into CYA mode. This became evident when a helicopter got shot at and the entire evacuation effort came to a screaming halt. Instead of counter attacking and immediately securing the surrounding area they instead sat on their hands until someone higher up could take over and therefore assume the blame for any further negative incidences.