Why use a relatively powerful chip for a simple task like this?
Economics. It's the same reason you'd send email spam, which requires a very high technology, instead of mass-mailing letters, which requires basically no technology. Computers and the internet, despite the high technology involved, are so ubiquitous that sending electronic spam is cheaper than actually mailing letters.
While email requires computers and the internet, which are each very expensive things in themselves, the incremental cost of sending an email is very nearly zero, once all the infrastructure is in place.
Sending a physical letter, on the other hand, requires a real human to carry a real letter printed on real paper and put it in a real mailbox. The cost of the paper alone makes the incremental cost of this method higher.
It's the same with a birthday card. Once the infrastructure is in place (semiconductor fabrication facilities, the IC masks, etc), the incremental cost of making that device is very low. Furthermore, the technology is such that the same device can play any number of songs, and in fact may be usable for applications beyond birthday cards. As such, the development costs are amortized to nearly zero, and the incremental costs are very low.