Also, some services, like, say, DNS servers need long-term portability of specific IPs across multiple providers.
I do both, actually on Vultr.
Unless they're TCP services. Or stateful UDP services. Then you're in for a world of pain if you try to anycast them.
The list of sensibly anycast services is surprisingly low. Stateless UDP services, essentially, which isn't a big list. Especially if you're the kind of person who's using Vultr's cloud, and not a megacorp.
If you have the resources to develop backend state sync, you're probably still better serviced with a (set of) load balancer(s).
I run a CDN with anycast BGP, this obviously uses TCP. Works great. I've also done it with a global network of physical datacenters and our own transit/peering network but eventually migated to the cloud for reasons.