I'm having a fundamental difficulty in seeing much utility to this particular type of API in relation to SIP trunks.
Do you regularly manage telephone systems for large corporations with hundreds of lines and thousands of phone numbers?
If not that might be why you don't see the advantages of such a system :-)
Is it really so time consuming to click the buttons?
Yes. Yes it is.
In the same way that it is time consuming to install, configure and deploy a server, which is why Amazon AWS is making a killing in the server market place providing a programmatic way to do this (Netflix deploy approx 10k servers in under a minute, run video conversion across all of the, and they pull them down again an hour later without anyone manually having to do anything).
It might not be time consuming to do it once at home to dial your mum, but that isn't the use case of Twillio. Twillio's customers include Walmart and Uber.
Why waste time doing something manually if it can be automated, particularly if it costs time and money for an employee to do it.