I found this thread trying to figure out why my own BrowserSource overlays weren't working - and thought I would take a look at the code in this plugin.
It has the same issue I was having - this plugin uses BroadcastChannels to communicate between an OBS Dock and a BrowserSource on a scene, however OBS27.2 has changed the way it technically implements BrowserSource elements if you select "local file"
Docks are added (as I would expect) using the file:/// scheme - (eg file:///c:/obs/file.html)
In previous versions of OBS, BrowserSource local files were added the same way - nice feature, BroadcastChannel works to communicate between the two windows
In OBS 27.0 - BrowserSources set to Local File add the file using an absolute/ scheme (non-standard afaik) (eg absolute/c:/obs/file.html)
This breaks BroadcastChannels - as they only work using the same scheme and host.
FIX / TLDR:
Change your BrowserSource from a Local Source to a manually entered file:/// uri, changed all windows back slashes in the file path (\) to standard uri forward slashes (/), and replaced any spaces with the uri encoded %20 for good measure.
eg Local Source c:\obs project\file.html becomes standard url file:///c:/obs%20project/file.html
I can confirm this Method works - Uncheck the Local File Checkbox - open the Browser-Source.html in the Animated-Lower-Thirds Folder Location in your browser and copy the entire URL. and Paste it in the URL location for your Lower Thirds Browser Source in OBS. Click Ok and your Lower Thirds should once again work...
Hope This Help you like it did me :D Thank you WoodCraftbyOwen!