Why doesn't a that-clause follow "in spite of"?

Solution 1:

Your question incorrectly presupposes two things:

1st presupposition:

A preposition such as of takes as complement only a noun phrase (a phrase headed by a noun).

Although a preposition typically takes a noun phrase, it doesn't always do. Here are some counterexamples:

a. get out of here

b. of late

c. in spite of me not having studied much for the exam

None of the emboldened complements of the preposition of is a noun phrase.

Now, if you're to argue that me not having studied much for the exam is indeed quite similar in function to a noun phrase, albeit not exactly a noun phrase, that's where your second incorrect presupposition comes in:

2nd presupposition:

A that-clause such as that I hadn't studied much for the exam is often called "a nominal clause" or even "a noun clause" simply because its function is considered to be similar to that of a noun phrase. Therefore, a that-clause must be allowed wherever a noun phrase is allowed.

Wrong!

Note that the emboldened word is similar, not identical. And the complement of a preposition is a function of a noun phrase, but not of that-clause.

If you hadn't presupposed these two things, I don't think you would have had any reason to ask the question in the first place.