I would understand if this was an edge case, but my understanding is this is a quite common and a core use case. I'm pretty sure this exact use case is in their basic tutorial.
If there is any possibility that those messages could be interpreted as spam, that doesn't seem to be a valid use case for Twilio (any more). You could forward the message over some channel other than SMS to the employee (like email, or a custom solution).
Also, I was looking into the pricing a few months ago and its kind of expensive because I'm actually paying for two text messages: One for the incoming one, and one for the forwarded one.
The code has been working this way since maybe 2015, possibly before (its been so long, I can't remember). I've been using Twilio since I think 2010. I started out just doing call forwarding and later on added text messaging when customers started to expect to be able to text message us.
At the time, it was a core use case. There was a tutorial and everything "Forward text messages to your phone in a few lines of code".