Sure, but it shouldn't be government operating on the frontier of new technology
Also, the US has been involved in AI research since the 1940s. So it’s not exactly a new thing.
Government has, practically and relatively speaking, infinite resources and a monopoly on violence. You can’t compete with it in any true sense.
Set a track record of Sherlocking private industry and you won’t have one. In countries that regularly nationalise and expropriate, the government (and its cronies) have to do basically all private investment and innovation, and resultingly, there tends to be very little of the latter.
The government was involved in basic internet research. It didnt try to operate pets.com