Update import search public channel browsing to use public API

Background

• The Import Search has a browse UX, which allows the user to browse their own channels or public channels in the content library
• The Import Search search UX was updated to load content node information from the public API, since we should use only the published state of the channels, available via the public API

Desired behavior

When browsing public channels, it should load the content nodes from the public API
https://github.com/learningequality/studio/blob/7700c2d5e94ccc22439c048b88db37642262e39a/contentcuration/contentcuration/frontend/channelEdit/views/ImportFromChannels/ContentTreeList.vue#L159-L164

Current behavior

It only filters the nodes to those that are published, but instead uses the current state of the channel's content nodes which could be different from the last published content

Value add

Continuing to align interactions with the content library with only the published state of the channels and they're contents

Possible tradeoffs

How should we split this behavior in the frontend components?

Start by reading ContentTreeList.vue around lines 159-164 and trace how browsing public channels currently obtains and filters content nodes. Identify the existing public API path used by Import Search, then verify that browsing reflects the channel's last published state rather than current unpublished nodes. Done means public-channel browsing loads its nodes from the public API.