"I do think moderate Muslims need to visibly and vocally take back their faith from the extremists, however. "
The problem with this is that in order to do this, they have to make a new "faith". Islam at its heart is NOT peaceful in any way, shape, or form. The founder of Islam was a documented murdering maniac - and that's from Islamic sources. Unlike Christianity, Islam spread by the sword from the very beginning.
I'm afraid the terrorists here are the one's who understand Islam. The prevalence of Islamic charities funding jihad is in accord with Mohammad's teaching that those who do not actively want to fight (most moderates) can fund those who do.
Now, it's true that not all Muslims are terrorists - but they are one religous revival away from it.
"It does have serious problems, and these need to be dealt with by Muslims, noone else."
They desperately need a Reformation, but there is one massive stumbling block to it. Christians, for instance, believe that the Bible is divinely-inspired, but written by men. Muslims believe the Koran was directly dictated by Allah. No textural criticism of it is allowed.
"That being said, not all Muslims are terrorists, or sharia-advocating extremists, and pretending that all Muslims are like that is only likely to swell the ranks of the extremists and make useful suggestions likely to be taken less seriously."
So, it's our fault their Koran tells them to kill us? I trust you don't intend that result from your argument above. Read Sura 9.
You've identified the major stumbling block correctly, and it is a major concern.