Use underscore prefix consistently for private JS members

[jotaen4tinypilot]

According to our coding conventions, all (supposed-to-be) private JS members should have an underscore prefix.

• I think we are fairly consistent in regards to naming methods here, but it would be good to double-check this anyway
• For fields, we have non-prefixed ones, e.g. in <remote-screen>. So we should check all components.
• The conventions around our common identifiers this.state and this.elements need to be updated to this._state and this._elements, and all code refactored.

We also have to do this in Pro.