They are just unique identifiers derived from a master key, it has nothing to do with cryptocurrencies.
Please realise these unique identifiers are valid addresses on the bitcoin blockchain.
So what if you don't store any master keys? you are still a centralized point of failure. When you say that a certain address has not been submitted before, why should I trust you? (since the database that counts how many times it has been submitted can be altered)
There are over two billion subkeys per master key, the website is based on the idea that your particular key has been manipulated is so unlikely that you can rationally trust the result.
put a tamper-proof device in the bottle and be done with it.
That is indeed an alternative solution.