There is no "public" here. There is only "government". It might do a better job than Google at this. It might not.
Just like every other government IT iniative.
Additionally, that would give the government direct access to a lot of sensitive information that people don’t really want them to have. While it probably has under-the-table access right now, do you really think having overt access is any better?
If the thought of having the government running a search engine, even at arms length, doesn’t leave you absolutely shitting yourself in terror, you have issues.