No, that's false
And yes, you can have a full range class D. Infact, there are many available for car audio. Even moreso for home audio.
Now, not all class D amps can be ran full range. We're not advocating you toss your Hifonics Brutus on your component set. But the class D amplifiers that can not operate well full range can't do so because they are design limited. It's not a limitation of the technology, however.