If you want to get a sense for what great math education looks like in 2014, you're blessed with an abundance of options.
Pick up the TERC Investigations curriculum, or the CME Project, or grab a copy of some of the NCTM journals. Or go check out some of the amazing work that educators are doing and sharing for free online. Shoot, go check out Christopher Danielson's blog, or Dan Meyer's blog, or Kate Nowak's blog, or Sam Shah's blog, or Nicora Placa's blog.
Now, you write that you try to show kids the "interesting" parts of math. I know what you mean -- things like knots or groups or puzzles and games, much in the spirit of Lockhart and those influenced by him. (I'm thinking of this article, in particular.)
What's remarkable about all the people that I linked to above is that they are innovating in education while teaching the same topics.
How you can help #1: Don't advocate for changing the curriculum.
This lets the entire system off the hook. The problem is not that school math is a different subject than real math, or at least it's not because the subjects that kids learn aren't genuinely mathematical.
School math is genuinely mathematical, or at least it should be. (The Common Core Standards say as much, by the way.) Every curriculum or person I linked to above excels at giving kids room to think mathematically about school topics. That means that rather than asking kids to memorize times tables, they ask kids to use strategies that make sense. Those educators use all their creativity and brain-power to figure out ways of presenting kids with opportunities to be mathematically creative with traditional topics.
How you can help #2 Use amazing resources for your students.
No educator is an island, and it's important that we don't all think through our teaching alone. I'm not saying that you are -- it sounds like you're a really thoughtful teacher! Still, it must be particularly difficult for you, as a tutor, to plan an activity or a lesson for your students. I would still recommend preparing yourself for your sessions as well as a teacher would prepare for class.
If your student is struggling with linear equations? Go do some reading about the struggles kids have with algebra. (May I recommend Fostering Algebraic Thinking?) If your student is struggling with rational functions? Go check out a high-quality textbook that's full of interesting problems and activities on the topic. (May I recommend CME Precalculus?)
How you can help #3 Show your students the multiplicity of mathematical talent.
It sounds like you're doing this already, but if school math often overemphasizes getting the right answer of remembering a procedure, then your students don't need you to do that. Instead, emphasize other mathematical talents. Value questions as much as answers. Value thinking as much as answers. Value ways of seeing, and not just ways of manipulating. Value explanations as much as answers. I'm sure others can add to this list.
Thanks for being such a thoughtful and supportive tutor! Your students are very lucky to have you.