Improvement
Start a new template from the picker instead of hunting for it
When no template fits, the picker now offers a New template button that takes you straight to the create page, including when a search comes back with nothing.

When no template fitted, you had to leave the picker and go to Message Templates to make one.
The picker now has a New template button, and it appears when a search comes back with nothing too. One click and you are on the create page.

Open the template picker when you are picking a template for a broadcast.