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

A good way to understand that without too much technical vocabulary is to remember that grammatical words such a "of" and "that" do have a purpose in a sentence; they are not content word, that is, they do not confer the idea of something or someone that is being talked about, but they indicate a relation between content words or between two parts of the sentence. The idea of introducing two such relations at once between two words or two parts is essentially foreign to the world of grammar, or, when that happens because of complexities beyond control, the result can be quite embarrassing for grammarians (this is exceptional, however). An example of that is coordination which is doubled of an apparent subordination suspected to be indicated by the coordinator; in this case the two relations are embodied by a unique grammatical word (and) but the problem is the same.

• Go by air, and save time. (conditional use of "and")

"Of" is a preposition and as all prepositions it must have a complement (always a noun phrase). Similarly "that" is a conjunction of subordination, and it must introduce a clause. So, you can have sentences as the following.

• He went to the beach in spite of [being sick/his sickness/…]. ("being sick" acts as a noun phrase.)

• He went to the beach in spite of the fact that he was sick.

In the second sentence you can use "fact" after "of" because "fact" is a grammatical unit that is proper (a noun), and also because the meaning of "fact" describes what is said by the clause introduced by "that"; finally, you can use "that" after "fact" because "fact" is a word that can be followed by "that" (You check whether that is possible in a dictionary). There are plenty of words with this property: reason, idea ("feelings" only), intention, …
For certain nouns a construction with "that" is not possible (most nouns): joy, freedom, anger, etc.
For others the two possibilities exist: intention, realization, etc. That way you do not have to make an adaptation by the addition of a word above so as to use "that" instead of simply "of" (but you might have to make other adaptations).

• The realization of the necessity of changes was immediate.
• The realization that changes were necessary was immediate.

However, as usual, you can't use both grammatical words together and leave one without the element that goes naturally with it.

that I hadn't studied much for the exam is a content clause. It is also a noun clause but its functions are limited compared to other noun clauses and simple nouns and noun phrases.

The subject is addressed at https://en.wikipedia.org/wiki/Content_clause

In grammar, a content clause is a subordinate clause that provides content implied or commented upon by its main clause. The term was coined by Danish linguist Otto Jespersen. They are also known as noun clauses.

It can operate as a subject: That I hadn't studied much for the exam worried my parents.

It can operate as the complement of a noun - as in your first example.

It cannot operate as the object of a preposition - in your example, "of"

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.