Are we down to a definitional rather than conceptional problem now? That's good -- understanding is just around the corner.
There's a difference between Islamists and Muslims -- that is why poli-sci folks spawned the term, "Islamists." Islamist is a political term, not a religious term. Expressing a goal of destroying Islam or destroying Muslims, or destroying Arab countries, or destroying Shia countries or destroying Suna countries is far, far different from expressing a goal of destroying "Islamists".
I think Christianity is the largest religion and perhaps Roman Catholicism alone is larger than Islam. But I don't give a damn. Islam embraces about a billion and a quarter to a billion and a half people around the world. Probably most of them are mad at the west in general and at the USA in particular. That's a problem, but not a problem of violence for us. It is a problem of political perception and political belief and economic disparity.
I'll bet that Al Qaeda and violent sympathizers worldwide have been degraded from a high of 10,000-15,000 (what's that, 0.0001%?) in 2001 to under 6,000 - 8,000 today (including replacements) and even that group is severely degraded in it's financial backing and internally within it's command and control. Part of the reason that it is difficult to detect, infiltrate and destroy is it's peculiar shared religious interpretation. It is a religious interpretation NOT shared by nearly every Muslim on the planet. It is not even shared by most of the people in the two regions where it is most common -- Arabia and the Hindu Kush.
If those mad bastards were NOT killing so many people (mostly Pakistani, Afghani, and various Arabs and Africans) nobody would give a damn about their religious interpretation and implementation of Islam. But this group does kill people for political goals, for religious enforcement and for a common religious vision of a world under their version of Islam. To ignore the salient shared characteristic of this group is to ignore the key to understanding the group. That is a foolish, foolish thing to do.
Who is ignoring it? And your post points out a very important characteristic. You say "that is why poli-sci folks spawned the term". Again, you use a Western definition to try and conceptualize an issue that is not Western in philosophy or logic. Simpler, OUR "poli-sci" guys defined a label for a group of Ismamists. Somewhat like a group of New Yorkers defining the "White Southern Male". Without being intrinsically involved with the culture it is nearly impossible to define either.
Back to Islamists. You suggest that we have to look at their religious aspect in order to understand them. Of course. But that shared characteristic of religion is not as relevant in a region that is pretty much 100% Muslim. If you had a diverse religious population [other than the various threads of Islam], with a population pool in which you could draw a conclusion that the only folks in the area that are dangerous are Islamist, you can start building a case that the "shared characteristic" is significant. But if 100% of your population is X and .00001% of X are bad guys, whatever X is [in this case religion] becomes insignificant.
So, back to the original point. When you use a term that Islamists understand to mean "people of Islam" and suggest that we destroy Islamists, that falls under the categories of of bigotry and racism because it suggests destroying the people who practice Islam.
Now about "But this group does kill people for political goals, for religious enforcement and for a common religious vision of a world under their version of Islam." With minimal editing that sentence could just as easily define sects of Christianity.