I think that the students who really understand trigonometry after taking a trigonometry course are only going to be the gifted ones. It's just a fact of the distribution of human talents -the same for algebra, chemistry etc. The less gifted ones pick up isolated facts, memorize simple patterns etc. I also think students are more likely to grasp and enjoy simple computer programming than algebra.Although I think the curriculum in high school is rather pointless, wasteful, and underchallenges many students, your proposal would be something for more gifted students and not for the norm.
If you want to teach people practical manual arithmetic , it's true that you can teach it as arithmetic in a given context, like figuring out interest on a loan. But I don't buy the argument that "everybody" must learn these practical contexts and I don't think that the students who don't remember the trigonometric identities will remember how to compute interest on a loan unless they do it regularly. (If you want to teach people how to figure interest on loans, you could start by forcing them to go into debt.)