On-page interaction does not require a templating language with logic. You can just as easily calculate the required values in your javascript and then pass the objects on to Mustache. That's a tangential point.
I fully agree with you that having some logic in a template language has its appeal. At first glance it sounds like a good idea (oh, I don't have to go back to the code to calculate that value, nice!) However, as I said, it's a slippery slope and depending on where you and your project end up, you may pay a dire price for that. In the end, inevitable, some part of your logic ends up in the template. (Been there, done that, been burnt)
Mustache/Handlebars allows for most of what you want - reuse things, split things into parts (templates can include other template) without including logic and thus creates a barrier that you cannot cross. It forces you to adhere to best practices.