Government programs related to btc or crypto will indeed raise awareness to people , hence, it can increase adoption as a result. With the government's backing, people will have more confidence in trusting this market. Most people are still hesitant up until today to get involve with btc because of the notion that this market is a scam or can easily screw their funds. But with the government directly participating in this market, it can give an assurance to people that this is not a scam after all. The technology is not a scam itself but it is the people handling crypto-related projects that are creating the bad reputation as some of them are really screwing people's funds.
Definitely Government involvement in crypto could significantly increase its use and much of people trust. Right now many people do not get involved because they worry about scams. When government supports or uses cryptocurrencies it makes them look more official and less risky. This also helps people see that technology itself is not problem it is people who create scams. Government programs could lead to better rules that protect people and build more confidence in market.