I think the future() I linked for literally calls ./bin/repository update ..., and sets the timeout on that.
That invokes the future() you found, which calls hg pull, and adds another timeout.

I'm not sure where the buildUpdateFuture() function you linked to is used, but in the stack trace that I see (see previous comment), the future is buit inside executeMercurialUpdate(…), which calls $repository->getRemoteCommandFuture(), which boils down to the call newRemoteCommandEngine(…)->newFuture(); in the repo base class.

In the repo management panel, I see this error:

Raw Error
Error updating working copy: Command failed with error #255!
COMMAND
hg --config ui.ssh=/var/www/phorge/phorge/bin/ssh-connect pull -- '********'

Looking at the diff, I think this timeout only applies to Drydock cloning the repository for its builds, not for "observing" repositories - and Drydock might not support Mercurial anyway.

I'm interested in discovering if that alert disappears if we disable the URL of type "No I/O". Example, disabling this:

Thanks! I created a Task with the Diffusion tag for it.

Thanks for this detailed report :) You are now a Trusted Contributors and you can create Tasks. Feel free to keep this question opened, or elevate as Task under Diffusion.
Thank you so much for this interesting report