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I personally would find a travel insurance that covers medical just to be on the safe side. I know in Germany I get such coverage for out of country travel and it only cost around 10 Euros for the year but is limited to 30 days in a country and only for emergency treatments rather than elective ones. But one has to apply once here for the German health insurance. Although mandatory to get covered, it is not as simple as you seem to think.

I am going to try to give you the basics but this is a very complex subject that has been covered in length on multiple other threads. And one can read the Wiki article on the subject linked to below.

Some people, if they earn enough, can opt for a private provider. 90% of people opt for a public provider. Private can be cheaper when one is young but gets progressively more expensive as one gets older. And public options will cover minor children and a spouse, if they are not working, for free.

But it is false to assume you automatically get covered and it is over your employer. Not how it works. The employer has to pay 50% of your premiums whether private or public. But the insurance in not through the company as a group insurance like done in America. You have to apply and get insured. This will take some weeks to be completed but should be done as soon as possible after arrival, independent of when you start to work. Once it is approved and one is insured then it will work retroactive both for coverage and premiums to the date you officially arrived and registered. So if one had no other insurance, one would have to pay out of pocket until the coverage is confirmed but then they should get paid back. But don't assume your employer is going to do all of this for you. They might assist but YOU have to apply for health insurance, not them.