A line of thinking that I haven't seen in the discussion about why conservatives are so eager for us to use the phrase "radical Islam" comes from my acquaintance with radical Christians. Don't worry, they aren't offended by the term.
You see, something I think many people outside of the evangelical movement may not realize is that “radical Christianity” is by no means a pejorative: it doesn't mean unusual except insofar as they regard true believers as a beseiged minority in an increasingly non-Christian world, and it connotes extremity only in that extremism in the pursuit of virtue is the very opposite of vice. It's an interpretive movement for modern evangelicals that embraces the “from the roots" meaning of the word. Many of these believers make the claim “Jesus was a radical.” Radical Christianity is, in their view, a life-suffusing, transformative relationship with Jesus, literally changing the believer from the roots. There's also some degree of marketing in the term, with "radical" playing a role similar to “eXtreme" or “turbo” in advertising the movement.
It has also been embraced by quite mainstream organizations. Here are a few websites from recent years that celebrate the term:
The Southern Baptist Convention: "Radical Christianity is Ordinary Christianity"
Patheos: "'Radical Christianity' vs. regular Christianity"
www.radicalchristianity.net: from the horse's mouth
The title of the SBC article is part of my main point. For these Christians, radical Christianity is the real Christianity. When you're fundamentalist, only fundamentalists are real adherents -- even if it happens that you're adhering fundamentally to something completely false. In fact, they would rather prefer it — they prefer their world divided neatly into enemies and friends. Hence their eagerness to hear government officials state that our enemy is “radical Islam”: because the immediate syllogism is that radical Islam is the real Islam, and hence our enemy is… Islam.
There may be other reasons that the Republican base is so eager to hear the phrase; there are other thought shortcuts by which the phrase “radical Islam” can be conveniently elided to suggest enmity with all of Islam, and for some conservatives it may simply have become a shibboleth. But I think it’s a heretofore overlooked ingredient in the mix.
(In my non-expert opinion, Omar Mateen was acting on a hatred that derives less from Islam than from a more generalized bigotry that happens to have expression in both Christianity and Islam. When reports suggest that he talked about being involved with both Al Qaeda (Sunni) and Hezbollah (Shiite), his own understanding of Islam and, indeed, political Islamism was probably none too clear. The presence of encouragement to terror did probably play a role in spurring him to action, but whether he would have done the same thing without such incitement is a question to which we can only reply with probabilities.)