The whole point of an open mail relay is the 3rd party can send their crap through your servers to anyone they want to target. This scenario gives the spammer no control over who gets the message.
Twilio is being used as a message bus, and OP is simply trying to save the work of writing an app or a new UI on the employee’s phone to read a list of messages. It’s a pretty common use case.
To the degree twilio's business model depends on allowing its customers to send to whomever they want (which I assert it does), they CAN NOT let those customers send spam - full stop - end of discussion. If you don't understand why this is critical to twilio - I don't know what to say. MOST users do not want SMS spam.
Seriously, if you need to send SMS spam, use one of the "bulletproof" SMS spam players.
Even if we say customer is not a spammer, they are sending mail out to a wide variety of third party networks. In email land a closed SMTP server only forwards messages to an internal network or server under that admins control. As soon as you start blasting spam out to third party systems you are not a closed relay. That's literally how the verizon and other carrier systems will see this source of SMS spam and phishing.
The point about the reputation of IP or carrier / telephony origintion numbers / space remains. The use case here is prohibited by twilio's terms of service. Maybe just comply with those or find another provider.
And I don't need to send sms spam. If it was blocked on the way out, that'd be fine.
A lot of HN has turned into - X company MUST build me Y feature so I can do Z. Actually, if you read the docs, they provide clear guidance not to do Z (for whatever their reasons are). They have the same issues with reputation that someone running a mail server sending mail to users across multiple networks and third party email systems might have.