Firstly, I am supportive of heavily subsidized/free post-secondary education. However, there is a substantial difference between K-12 and post-secondary.
K-12 is considered a baseline for society. Everyone is expected to have it. It absolutely makes sense for the government to pay for it.
Post-secondary is a different beast. It is not a baseline for society (Even if it is a baseline for white collar-work). Not everyone is expected to have it. People who end up in, on average, better-paying jobs are the primary beneficiaries of cheap/free post-secondary education.
There are two ways out of this situation.
1. You can make post-secondary the baseline. Everyone could be expected to have it. This... Is probably a good idea for building better citizens (and, by proxy, a better society), but it is not economically cost-optimal.
2. You can keep post-secondary as a special privilege for the a select few, citing that society, as a whole, benefits from educating its most capable members. In that case, you'd need hard admissions testing/filtering/etc, it should be easy to drop out if you can't perform once you've been admitted, and universities should primarily focus on education, rather than being a four-year, three-star resort for middle-class young people.
Neither of these two options would look anything like "College, as it is currently practiced in the United States, but paid for by the government."