As to your ultimate question, no there is close to zero chance that the German government will permit LH to become insolvent. Moreover, at the time the first series of issues arose, the Covid-19 issue was growing but not thought to deeply affect Western Europe. No reason to believe that LH would be in poor condition.

Your mistake was in not turning this over to your card issuer (bank) for a chargeback after the first 7 days passed. Depending on where you are located, the time for filing a chargeback may have passed (but you should know that most significant US card issuers permit as long as a year even though only legally required to provide 60 days).

In your situation, start the chargeback process now. It takes 2-3 minutes to do this online and it puts the shoe on the other foot. As LH has acknowledged the refund being due, it is unlikely to waste time contesting your dispute .

For the benefit of others, time is money. Don't spend it on the phone. Give a reasonable period of time (in this case the 7 days LH said it would take is reasonable) and then start the chargeback process.