Edit to address cas's answer
My goal is not bleeding edge, but control over debug versions. I can't find a debug (non-optimized) version of libstdc++ (obtained by configuring GCC with --enable-libstdcxx-debug), on the Fedora repos. If I had this on a package, I wouldn't need to build GCC from source.
And, even though learning more about the build process and its interaction with the system components is a bonus, what I really want is to experiment with GDB, symbol-loading, and debugging optimized vs. non-optimized code. That's why I'd like to have the non-optimized versions of the libraries.
I agree on the trouble. My previous 2 attempts, where I followed the "normal" way you mention, have resulted in unusual errors building Qt (the most complex software I expect to build from source); and, when building with the "system toolchain", everything went fine. Which is why I thought that maybe there was a better way to do I want, more thorough, but without requiring building the whole system from source.
I can move to a different distro, provided I don't have to build everything from source, and I have no requirement on software versions, i.e., I could build GCC 5.1.1, which is the same I have on my Fedora VM, to minimize the differences. I'm already following the "build recipes" from Fedora packages as much as possible.