How do I get podcasts on my Apple Watch without needing my iPhone?

Correct, the link you have provided about Play podcasts on Apple Watch – Apple Support does not mention this.

But the previous page about Adding podcasts to Apple Watch, has the information:

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When you follow shows and create stations in the Podcasts app on your iPhone, recent episodes of those shows are available to download to your Apple Watch when it’s connected to power.

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Force restarting your iPhone and Apple Watch may also help to get the sync process started.

Force restart iPhone - Apple Support

To force restart your watch, press and hold the Digital Crown and the Side button together for about 10sec, until the Apple logo comes back again.

Can you sync other content to the watch, for example music?

Also make sure that the watch and the iPhone are connected to the same Wi-Fi and nothing is blocked, and restart your Wi-Fi router, just to be sure.

Sorry, but the Apple Watch User Guide for watchOS 11 mentions, that the watch has to be charged during syncing podcasts:

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When you follow shows and create stations in the Podcasts app on your iPhone, recent episodes of those shows are available to download to your Apple Watch when it’s connected to power. See Follow your favorite podcasts and Organize your podcast library in the iPhone User Guide for more information about following podcasts and creating stations.

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copied from: Add podcasts to Apple Watch - Apple Support

And, as far as I remember, every type of content like music, podcasts, photos, etc, can only take a certain amount of free space on your watch. So you will not be able to use the total free space on your watch to download your podcasts.

marc cardwell wrote:

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How does Apple decide what music and which photos get copied to the watch? I have over 3gb of music, and 20 mb of photos and, allegedly, 27 mb of podcasts.

You can choose which music you want to add to your watch, or let Apple select the music for you.

Using the Watch App on the iPhone is one option, selecting the music directly on the watch is the other.

Have a look at the steps mentioned in the Apple Watch User guide:

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You can add specific playlists and albums to your Apple Watch using the Apple Watch app on your iPhone. If you’re an Apple Music subscriber, you can also add music with the Music app directly on your Apple Watch.

You don’t have to choose specific music to add to your Apple Watch if you’re an Apple Music subscriber. Music you’ve recently listened to is added automatically. (If you haven’t listened to anything, music recommended by Apple Music is added.)

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copied from: Add music to Apple Watch - Apple Support

Selecting photos works in a similar way:

Choose a photo album and manage storage on Apple Watch - Apple Support