Watch: System media player sometimes does not detect playback from Pocket Casts watch app

### Description

When playing audio through the Pocket Casts watch app, the system should detect that playback and reflect it with an ongoing activity in three ways:

1. The watch's home screen should have a little animated equalizer at the bottom with the other notification icons
2. The watch's "Media Controls" app should be able to control playback
3. The Pocket Casts app tile in the list of app's should indicate the title of the playing episode

This usually works fine, but there are rare occasions when it does not work. It's not clear to me why that is.

### Step-by-step reproduction instructions

These steps to reproduce only occassionally show the problem.
1. Start playing an episode with the Pocket Casts watch app
2. Go to the watch's home screen and there should be an animated equalizer on the bottom of the screen
4. Go to the apps on the watch and open "Media Controls" and playback should be reflected.
5. Tapping the pause/play icon should pause or resume Pocket Casts playback.

### Screenshots or screen recording

This is a video of what it looks like when things are working as expected:

https://github.com/Automattic/pocket-casts-android/assets/4656348/8e7d7846-a2a2-40a2-83b0-e229712dc1a3

And this is a video I took this morning when I started playback and noticed the media notification wasn't working like I'm describing in this issue:

https://github.com/Automattic/pocket-casts-android/assets/4656348/2dfc9c1a-8e5d-4381-8742-882ff452fdb7

### Did you search for existing bug reports?

- [X] I have searched for existing bug reports.

### Device, Operating system, and Pocket Casts app version

Google Pixel 4, Wear OS 4 Build TWD4.231005.002 System UI 1.7.37.587865003, Pocket Casts 7.54-rc-1w,

No files or tests are named. Start by reproducing the intermittent failure on the listed Pixel 4 and Wear OS setup, comparing the working and failing recordings while tracing the Pocket Casts watch app's system media playback integration. Done means playback is consistently reflected by the home-screen equalizer, Media Controls, and Pocket Casts app tile, including pause and resume control.

Written by the indexing model from the issue text.