The best AWS service for sending SMS via REST API calls

SNS is the right service where you can make an API call to publish a message to a topic and subscribers to that topic can be SMS clients. However, you will need to go through a sandbox phase before getting full access to the service. See Mobile text messaging (SMS).

Hi.

You can use SNS to send SMS messages to unidentified users.
There is a caution. The following pages provide notes on sending unsolicited messages within best practices. Please comply with this.

https://docs.aws.amazon.com/ja_jp/sns/latest/dg/channels-sms-best-practices.html

And how do you manage thousands of phone numbers?
In addition to simply sending, you may be able to take advantage of Amazon Pinpoint for scenarios that require segmentation or workflow.