If a person is in 4 teams, and they're not a full-time 'product owner' where going to meetings is their whole job, I'd say something bizarre is going on.
My team has been an overlay team for years. We provide services to many teams. Since the glorious transition to agile and product focus, we had to sacrifice someone to be the product owner, so their 'whole job' is overhead management, which they hate. Now, every team wants someone from my team in every standup ("but we'll be blocked if a question for your team comes up and no one is standing in the corner patiently waiting to answer it") and definitely wants someone in every sprint planning. 8-ish person team, dozens of projects, and time disappears very, very fast. I've managed to push back on the 'every standup' nonsense so we don't spend 100% of our time in meetings, and in return the agile folks are using their 'metrics' to basically tut-tut and say we don't do any work.
I can see where if your whole job is on a single project where there workstream can be easily decomposed into 'sprint-size' chunks, agile and the rest are probably useful. But that's not every job, and maybe not most jobs, and it sure as hell isn't my job. But like so many other have concurred, and as I see on a routine basis, there will be someone with 'Agile' or 'Coach' or 'Product' in their job title will have an expansive explanation why I'm Doing It WRONG. Capital One seems to have figured it out...maybe there's hope[1].
[1] https://www.reuters.com/technology/capital-one-scraps-1100-t...
In agile you're supposed to know what's going into your sprint, and each ticket should meet your "definition of ready". Teams shouldn't need someone on hand because they should already have refined and unblocked issues before the ticket was added to the sprint.
Whatever it is you're doing it isn't agile.