Are the sets of a play crucial to creating the mood of the scene, or are they unnecessary?

Yes they are crucial for setting the mood before the action, but they are not as necessary once action on stage takes place.

In other words, if you want to set a mood or tone for the audience before the action begins, then the sets and lighting will certainly assist with the perspective. But on a bare stage, actors themselves can set the tone once the action starts- but not as effectively. Costumes, lighting, sets all play a part on stage and lend to the authenticity of the mood.

But why not use all the tools at one's disposal?

Mr(merely a player)Sensitive