I was talking about Muslims in Europe more than those in the Middle-East, can't claim any personal experience with that latter group unlike you. Although having studied Arabic, Islam and Islamic history, I think I have a reasonably good idea, albeit on a more academic than personal level, of the bizarre blend you describe. For sure, the most notable thing about opinion polls among Muslims like the Pew one published a couple of weeks ago, is how many times the results seem to contradict each other, at least at first glance and in the eyes of westerners, and the massive confusion about what precisely shariah means is only one small part of that.
Certainly you are right about how we shouldn't mistake ignorance for radicalism. But plenty of Middle Easterners, whether in Europe or there, are far from ignorant - their seemingly radical stances are born of the narrow intertwining of religion and politics that has characterized mainstream Sunni Islam almost since its beginning (and Twelver Shi'a as well, really), the way their leaders abuse both for propaganda to make people overlook their constant failures - and simply their perspective. I think it's important to understand how a Muslim can e.g. support Hamas or Hizbullah, without being either ignorant or particularly radical. Hell, even some non-Muslims do that, the more extreme elements of the pro-Palestinians on the European left (though one could justifiably ask why I call them extreme if they aren't Muslim, and not if they hold the exact same view but are Muslim...).
Don't misunderstand me - I am not saying the MCB made the right decision or the only possible decision there, necessarily, just that it's an understandable decision.
And I hope it's clear by now how I'm not a fan of Yusuf Al Qaradawi. But I will admit, my reasons for that have very little to do with his support for Hamas or Hizbullah. More with his arrogance and the way he's constantly referred to as such a big authority without ever having shown anything in the way of vision or leadership that I've seen - stop the presses, Qaradawi says the Taliban shouldn't blow up the Bamiyan Buddhas, then surely they won't! And being Egyptian, he gave some revoltingly weaselly comment on female genital mutilation in the article Hyoga linked to, which is actually a good illustration of how politics and culture and religion are inextricably linked, since e.g. an Iraqi or Iranian counterpart would have strongly discouraged or outright forbidden the practice.
Anyway - it's been a long day, and I'm not really sure how much sense I'm making here - at the end of the day Qaradawi IS one of the single most influential Muslim clerics in the world, and while the MCB could of course have included some obligatory criticism of him in their published statement, it makes plenty of sense to me that they didn't, and I don't think that makes them radical. It just makes them tightrope walkers, who choose to conserve political capital for battles that matter more. Radicalizing young Muslims already find them weaklings who kowtow to the political establishment, I've no doubt, and would think that even more if they criticized even someone as mainstream as Qaradawi.
In (Arab) Islam and in Arab politics both, there is a great deal of "political correctness" as I said - so many things one has to pay lipservice to, so many things one can't publicly say without being attacked from all sides. But one has to realize that, and realize that due to that, people may not always be as radical as they seem. Some of the things they say may just be formulaic and near-obligatory without them really meaning them. Arab politicians in the West, in which group I include the leaders of organizations like MCB or CAIR, have real and concrete goals and causes that they need their influence and their political capital for, such as trying to keep radicalizing youth under control, and so taking on those taboos is really just a stupid distraction from that.
Note: Yes, I'm aware that there are more Pakistani and Indian British Muslims than Arab ones, and so the matter gets more complicated, but let's not go into that.