The Scrum Master is a coach and facilitator, as well as a process referee. There's no reason a Scrum Master couldn't or shouldn't facilitate team building exercises, but when you start asking "Whose job is it to make lunch reservations?" then you're already past the point where going to lunch together is likely to solve anything.
Analysis and Recommendations
[I]s the Scrum master in charge of organizing [a] team lunch, finding restaurant (sic) etc?
This is an X/Y problem. While the 2020 edition of the Scrum Guide says:
The Scrum Master is accountable for the Scrum Team’s effectiveness.
and includes a specific bullet point making the Scrum Master responsible for:
Ensuring that all Scrum events take place and are positive, productive, and kept within the timebox.
you are fundamentally addressing the problem the wrong way around. Your problem isn't about whether you go to lunch, where you go to lunch, or whose job it is to plan a lunch; your real problem is that the Scrum Team (including the Scrum Master) isn't behaving like a cohesive team.
Deeper Analysis and Root Causes
Another way to look at this is that the Scrum Master is a member of the Scrum Team, and as such can take on tasks that are useful in training, coaching, facilitating the Scrum Team's activities, or removing impediments to progress. If the Scrum Team agrees that team lunches or other team-building exercises have value, then there's no reason a Scrum Master shouldn't do the legwork while the rest of the team is working on other things.
However, since you wrote that "the team has bigger problems to solve" then it's rather dubious to think that taking everyone out for pizza or wall-climbing is going to solve the underlying dynamics within the team. These are issues that should be addressed within Sprint Retrospectives, additional team and one-on-one coaching during each Sprint, and generally addressed head-on.
A "team" isn't a team just because you slap a bunch of people together in the same room; a team is a group of people who are actively working together to reach a common goal. If you don't have that, you don't have a Scrum Team. Ispo facto, debating lunch planning responsibilities means that:
- Communications and collaboration within the team are problematic at best.
- You are well past the point where going to lunch is sufficient to solve the collaboration and interpersonal problems you already know you have.
- Is unlikely to uncover additional issues you don't know you have.
- You as the Scrum Master are also part of the problem, because you're one of the people debating whether something is your responsibility or not rather than collaborating with the rest of the team to get it done.
That last bullet is especially important. While a one-off lunch isn't going to make a big difference, the fact that you're even questioning whether a trivial responsibility is yours or not points to a core dysfunction within the Scrum Team.
My near-term recommendation is that you work with the rest of the Scrum Team to identify as many of the current dysfunctions as you can, and then work together to find a solution to the real problems the team is facing. If you need help and support to do that, then speak to your leadership team and ask for help and guidance from line management, your agile leadership team, and your company's agile coaches. If necessary, ask for additional budget to hire or contract an agile coach to help you and the rest of the team improve the Scrum Team's collaborative processes and its team members' ability to leverage the Scrum framework.