With BGP (especially anycast) you don't have direct influence where the requests lands. You can steer traffic by techniques like AS prepend, some priorities per BGP session (I'm not an expert in BGP) but eventually it's not CDN that decides where the request will be routed. It's decided by routers of the client's ISP and backbone networks, each taking decision themself (shorted AS path from his point of view + BGP policies set by network operators), not by you. You can't for example split traffic between DCs with specific proportion (like 30% here and 70% there). You can't split by something else than network properties like forward request to the region that has better chance to have this content in the cache.

With DNS and dynamic responses you are directing request to specific DC, even server, almost on every request. It may be dedicated for this traffic type (live stream different than static images etc). Your DNS server can take the hostname ("www.google.com") into consideration - BGP doesn't even know the hostname in the URL. If you wanted to do it with BGP you would need to place specific content to a dedicated /24 subnet and that is impossible considering how many IPv4 addresses are available.

BGP doesn't even consider network latency, current network load. CDN knows load on their machines, on their network link, where given content is placed. The bottleneck may be storage, network or CPU processing, different for different sites and content type. They need to direct traffic on request basis considering this and at least the hostname from the URL. That's why DNS is used first.