First of all, blockchain has nothing to do with mixing and anonymization.
A blockchain is a distributed ledger with growing lists of records (blocks) that are securely linked together via cryptographic hashes. [Source 1]. Blockchain technology alone (also) is very interesting and beneficial for everyone, for governments too. Some governments and companies invest into development of blockchain technologies in their countries, in different niche.
And yes, even without privacy, Bitcoin and blockchains would still be useful, just as these are still very useful today despite privacy being very minimal.
If privacy was important for people, Monero would be number 1 cryptocurrency on coinmarketcap instead of being placed at 47th place right now (with 24h volume - 136th.