That's a problem with bundling services under a single account, not with the closed-ecosystem part. For example, GoDaddy suspends all your sites when you get a DMCA takedown to any single one of them.
I agree that the approach is bonkers, but it's not an issue exclusive to "walled gardens".
If you were on the open web, his app would be a page with youtube's embed code for each video.
I doubt you would use a DMCA takedown request for this. You'd just disable embedding of your videos in your youtube options. Youtube probably already knows when a video isn't requested from either youtube's website or the official youtube app.
I'm not saying it is a good thing. I'm saying it's not a problem caused by "closed app eco-systems". Trademarks apply to both closed and open marketplaces.