The Traffic Manager internal architecture is resilient to the failure of any single Azure region. So, even if a region fails, Traffic Manager should stay up. That applies to all Traffic Manager components: control plane, endpoint monitoring, and DNS name servers.
Since Traffic Manager works at the DNS level, it doesn't have an 'endpoint' that proxies your traffic--it uses DNS to direct clients to the appropriate endpoint, and clients then connect to those endpoints directly. Thus, an unreachable endpoint is an application problem, not a Traffic Manager problem.
That said, if the Traffic Manager DNS name servers are down, you have a serious problem. You DNS resolution path will fail and your customers will be impacted. The only solution is to either accept the risk (small, but can never be zero) or have a plan in place to use another DNS system, either in parallel or failover. This is not a limitation of Traffic Manager; you could say the same about any DNS-based traffic management system.
The earlier answer from DornaDigital is very good (other than the first point which suggests DNS caching will protect you through a name server outage--it won't). It covers some important points. In short, DNS-based failover works well for new sessions. Existing clients may have to refresh or even close their browser and reconnect.