I think you may be confusing universal healthcare with socialized medicine, it's a common mistake.

Residents pay tax (insurance) into a common pot, and get universal healthcare. The government pitches in with contributions from the general budget. How is that not socialized healthcare?

Universal healthcare doesn't always work that way though, it's just basically the idea that every citizen is guaranteed some minimum standard of coverage. For example in Japan there's a mandate that all residents are required to have health insurance, but most people have it through their employers. It's not single payer or socialized healthcare. In many countries there's a mix of private and public and a portion of the population is covered by a system like you're talking about, like in the US with Medicare and Medicaid.