Beaucoup de refers to a large quantity. In your context, it can be replaced by Un grand nombre de.

But you are wrong when saying Beaucoup des does not exist.

Beaucoup des miens habitent encore dans les oasis.

Beaucoup des nôtres sont allés travailler à l’étranger.

Examples source

You can use beaucoup des when you talk about all the beings or things we talk about.

J’ai attrapé beaucoup de poissons.

Beaucoup des poissons que j’ai attrapés sont encore petits.

Before a personal pronoun, you can use beaucoup de

J’ai beaucoup de chance.

That’s a great question!

As a native French speaker, I actually never thought about it and I guess I just learned all of the special cases by heart without noticing. However, from your perspective, I can totally understand that it seems a bit confusing.

I don’t have any formal answer to offer you, but I can maybe give you a good heuristic to help you understand in which situations you should use each type of expressions.

Here we go:

The expressions beaucoup de.., énormément de.. or un peu de.. are used for the times you would say a lot of.., a bit of.., tons of.. in English.

However, de can become du or des when, in English, you would use the instead of of, like in beside the cat -> à côté du chat.

So as a rule of thumb, in a lot of contexts, of = de, and "de" becomes invariant, but every time you use "the", to translate in French, your de can become de, du, des, de la.

I hope this gives you a bit of intuition about when to use each form.

The only rule is that de combines with the articles le and les.

Normal articles

If an article is present, de will combine with it:

Les ours mangent. The bears are eating.

Des ours mangent. Some bears are eating.

Beaucoup de

This is true after beaucoup as well. But beaucoup with and without an article mean different things:

Je vois beaucoup de lapins. I see a lot of rabbits.

Je vois beaucoup des lapins. I see a lot of the rabbits (that we were talking about earlier).

Apostrophe combinations

Also, apostrophe combinations always take precedence over de combinations. Here we get du :

Le temps est venu. The time has come.

Beaucoup du temps. Much of the time.

But here we don’t, because the l’ is more important:

L’ami de Marie parle beaucoup. Marie’s friend talks a lot.

Je parle de l‘ami de Marie. I’m talking about Marie’s friend.

Direct objects

There is one common kind of sentence where you don’t get a combination. It’s when le and les aren’t articles, but direct objects of a verb: le as in “him” or “it” and les as in “them”.

A-t-il déménagé ? Did he move away?

Non, je continue de le voir de temps en temps. No, I continue to see him from time to time.

Les enfants sont-ils bien contents ? Are the children quite happy?

Oui, leur grand-mère s’efforce de les gâter ! Yes, their grandma is doing her best to spoil them!

à + le, les

Luckily, all of the above rules also apply to à combinations.