The few protections it does offer against tampering can be achieved in a much simpler way for your scenario (signing)
Who cares about proving how physically access-restricted your server room is, any funny business will be noticed right away.
People holding cryptocurrency would notice an illegitimate takeover of the blockchain right away because they'd be trying to spend cryptocoin which, all of a sudden, they no longer have. But regulators aren't trying to tally up business inventory on their own ledger so that they can send it off to other parts of the business and all of a sudden that kind of logistics fails for the regulator because of what you called "funny business". A regulator is a passive observer, and a passive observer can't detect funny business without actively auditing the blockchain against their perceived notion of whether the current state of the blockchain is normative... which is a very difficult problem indeed to do at scale, one which regulators today haven't yet been able to really automate, even with the relative certainty of a database (which a regulator could order regular dumps of, for analysis, if it wanted to).
- Your company handles dangerous or highly regulated materials (prescription drugs, hazardous materials, etc.) and you are required to have controls in place to monitor your supply chain. - You deal with lots of vendors of questionable reputation, or have a history of graft, embezzlement, or other loss in your supply chain. For example, easily "misplaced" goods like cigarettes that need to be distributed to lots of retail locations.
So say you've got central logging setup at your organization. You're smart and are using rsyslog with SSL/TLS and your own CA. For the most part you can reasonably claim that your log messages are secure from the server that emitted them to the destination in your central logging system but can you guarantee they're not modified after that? No. You can't.
From this perspective using a blockchain for logging security-critical events would be extremely useful. It would be impossible for an attacker to modify the logs after-the-fact so that you could no longer determine which account they used to login.
You wouldn't want to use it for general logging because of the overhead but for things like login events, reboots, etc it would be fantastic.