Fun and crazy English is everywhere. One way to think about the real issue here is in the use of the ever-so-useful verb, "to have." Leaving aside its powerful usage in stretching out time in the 'perfect tenses,' its meaning of ownership is one that people use constantly: "I have a pain; I have an idea; I have a problem."
The use is always transitive. A direct object is required. Even the response, "I have!" to a question like, "Who has a sandwich today?" implies the direct object that the question is inquiring about.
And of course every direct object has to be a noun. This is one of those places where the flexibility of English phrasing is pretty cool and also pretty weird. Infinitives are the 'perfect' form of a verb, but they can never, ever be a verb in a sentence. Instead they are, much more often than not, phrases that do the work of a noun. "To fight is dangerous; to die today would be horrible; to play with friends is fantastic."
That is always what is happening in sentences like "he has to be sick," or in complete thoughts like "every direct object has to be a noun." Perhaps some time in the past, English users thought of their infinitives more clearly as nouns.
Perhaps sometime in the future, some grammarian will say, "We'll just call 'has to' a phrasal verb." If that ever happens, trouble will follow. How can your direct object be a verb phrase?