Governments have been in recent decades completely hands off from anything tech related. For the longest time they followed the usual neoliberal economist trope that markets solve everything optimally. As a result they have created in the so-called "big tech" conglomerates a monster fit for the history books, the most broken and cornered market there ever was.

But a wholly government run search engine is not a solution. There are inherent biases in both constructing and presenting indices. You dont want to further stoke the anti-commons mistrust of polarized societies.

What the public sector could do is fund all the background techonologies to make it easy to have much larger numbers of search engines. Some of those assembled services could be completely open source, others could be value adding with various added services and customizations.

In any case the status quo is a disaster that has no future. Its effectively a forced dumbification of society as it artificially suppresses the flow of high-quality public information. Incidentally it also doesnt solve the problem that much of the world's information is private. Desktop search should become a thing again, in line with local AI etc.