The bigger problem, and where BGP multihoming is most handy, is it's just so much easier to get a holistic in+out failover where nothing really changes vs in DNS where it's more about getting the future inbound stuff to change where it goes. E.g. it's a pain to break an active session because the address had to change, even if DNS can update where the new service is quickly.
Using the wrong route to get the packet in your general direction still gets you the packet as long as it hits an ISP along the way that got the update.
We could fully drain traffic from a transit provider in <60s with a withdrawal with all of the major providers you get at the internet exchanges. If you weren’t seeing that your upstream ISPs may have penalized you for flapping too much and put in explicit delays.