Perhaps your employee's cell carrier detected the message as spam and automatically complained to what it saw as the source (twillio). From twillio's point of view they absolutely and unequivocally need to keep the carriers happy. If Att or Vzw simply blocks all twillio then twillio is dead. When faced with the choice of losing the entire company or losing your $600 a month it's pretty obvious why twillio has in place the automated system you recently chafed under.

It's probably a good idea for twillio to be more reachable to any paying customer, at least until a human employee makes the determination if the customer is a spammer.