Global load balancing: GeoDNS vs Anycast vs application-level routing

We're looking to expand our services to Asia and Europe, and global load balancing is obviously key to keeping latency down for international users. We're evaluating different approaches: GeoDNS, Anycast, and application-level routing. Each seems to have its own set of tradeoffs. GeoDNS is relatively easy to set up but has caching issues with local DNS resolvers, meaning users might not always be routed to the closest data center. Anycast looks promising for truly directing traffic to the nearest PoP, but it requires BGP expertise, which is a bit outside our team's current wheelhouse. Application-level routing, where we route users based on their IP after they hit a central entry point, adds an extra hop and might not be as fast as routing at the network level. We also need to consider robust failover strategies: what happens if an entire region goes down? What approaches have you found most effective for global traffic distribution and resilience?