Is it appropriate to email a professor with a simpler solution than the one she provided?
Email the professor, explain that you came up with an alternate proof, and ask if it is correct.
If you approach it this way, it should be perfectly acceptable to anyone.  You get to find out if your more-clear approach is correct, and assuming it is, the professor maybe uses it in the future.  If it's not correct, you get to learn where you went wrong.  A good exercise for all involved.