Quite a lot / quite a few / quite a bit

A few only with countable nouns. Generally a low quantity. A few people helped me finish.

A bit of only with uncountable nouns. Low quantity. A bit of help goes a long way

A lot of with either countable or uncountable nouns. High quantity. A lot of people know me!

Now by adding quite to these, they reduce the idea of a lot, and they increase the ideas of a few and a bit.

Quite a few people helped me finish. Now it's a good quantity. Quite a bit of help goes a long way. Now also a good quantity. Quite a lot of people know me. Now this is not as emphasized as before, it has become a little softer.

Hope this helps!

Regarding the fact that I agree with you about the same meaning of quite a <COUNT-WORD> and many. I think these there have the exact same meaning but they better be used when they fit in the context of the sentence although they can be replace each other pretty freely. For example, I

• It's better to say "He wrote quite a few words about recent incident in UK lately." than using "quite a bit of words".
• Or "He has quite a lot of knowledge in this field" rather than using quite a few knowledge or better to rephrase your sentence if you like to use it like: "He knows quite a few facts about this subject".
• Or "There is quite a bit of information in this book listed about the theory of Compute Science" is make more sense than using the others.

So, the only thing that I can emphasize is to use a few with countable word and try to pick the best that fits to the context of the sentence.

Also, I would probably say "There are quite a lot of words on the page", but "There are quite a few words on the page" also sounds correct. However, "quite a bit" doesn't work in the same context.