The antecedent of a relative pronoun may depend on things other than proximity. Number, for instance. If I say
The Rockies is a range of mountains that is very large
I'm talking about a group of mountains, and that grouping covers a large area. The mountains themselves may be large or not. Range is singular, and so is the verb (is) of the relative clause. Thus the antecedent of that must be range. On the other hand, if I say
The Rockies is a range of mountains that are very large
I'm talking about a group of sizable mountains. The plural verb are must agree with the plural noun mountains, so the antecedent of that must be mountains.
Calling a group of sick people a sick group is acceptable, certainly in extemporaneous speech, so I wouldn't call Trump's statement ungrammatical.