No they don't. The onus on them is to comply with the takedown request and provide for immediate restoration with a counter-notice. If the initiator is acting in bad faith it will be exposed if they attempt to litigate.
The friction free restoration flow is what Google is missing because they don't actually follow the DMCA process. Amend the law to strip safe harbor immunity in this scenario and suddenly we'd see abuse effectively combated.
Etsy seems to operate in the same way as google. I had a DMCA against a listing of mine that should have been protected as parody. Etsy immediately complied with the takedown, and then I emailed the complainant and they agreed it was a mistake from their system/consultant. But they never took the next step of contacting Etsy to say they were wrong. So I could never restore the listing.