Use a separate logger for dead links check that Sentry will ignore

[dhruvkb]

Description

This PR changes the logger used inside the _head function in check_dead_links to a new one that will be ignored in Sentry.

Testing Instructions

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[obulat]

LGTM! Thank you for fixing this. I'd like to deploy it today to stop the alerts.

[dhruvkb]

Looks like we only require one review to merge now.