The government has successfully solved many large-scale infrastructure problems. Have you ever driven somewhere on a road? Was that road funded by you paying for a "road subscription" or as a benefit from your employer? Probably not. The government built it, and there are a lot of them.

I don't see why healthcare is any different. People wanted roads, so we built roads. People want healthcare, so we should build healthcare. It's the same exact thing.

Yes, it's called Medicare.

Medicare is expensive per capita, and also is not self-sustaining: it needs people paying and not getting benefits to subsist. The entire social security paycheck wouldn't cover the cost of medicare per capita.