Is ‘facilitating events’ service to PO?

It is also listed as a service to the Development Team. 

I think you have a fair point, that this could go into a service to the organization. Specifically the Sprint Review is an event where the wider organization can directly benefit from good facilitation.

I presume the reasoning behind this is that the events are run by the Scrum Team, and while a Product Owner will invite attendees, the Scrum Master may be the one who helps find a good time or location, and prepares stakeholders to be able to participate effectively. Similarly, the Daily Scrum is for the Development Team, but the Scrum Master may need to assist in multiple ways, such as ensuring the event achieves its desired outcome, that the timebox is respected, etc.

I probably see this differently than others. I see my role as facilitator not as someone who does it but as someone who helps it to occur and be beneficial. Facilitation is mentioned as a service to the PO and Dev Team.  As a Scrum Master you are helping those two roles to appreciate the events and to get the most out of their intended purpose.

I wait for the Scrum Team to ask me to facilitate but I do not lead any events.  I do not take notes in any events. Many of the events, I do not even participate even though I attend.  I am there to help.  Each event has specific purposes outlined in the Scrum Guide.  I facilitate the understanding and accomplishment of that purpose. I have teams that run all of their events without me and I attend to observe and suggest future improvements.

I think that the reason the Guide does not classify this as a service to the organization because if the Scrum Master provides the service directly to the PO and Dev Team.  It is indirectly a service to the organization.