Registrar doesn't host your NS records, they are hosted on name servers for a particular TLD your domain belongs to. So, if registrar goes down - absolutely not

Registrar doesn't host your NS records, they are hosted on name servers for a particular TLD your domain belongs to. So, if registrar goes down - absolutely nothing will happen. The only way registrar could mess things up is by updating your records incorrectly. And these things do happen from time to time, but are very uncommon. For example, I use domains for decades and was never affected by it.