[foobarqux]

For information theoretic entropy (I don’t know anything about thermodynamics):

The first thing you describe, that if you draw many symbols from your source distribution at random then you see a distribution of symbols that is equal to the probability distribution of the source, is called the asymptotic equipartition principle.

I think your confusion comes from two things. First, conflating bits and symbols and second assuming symbols are equiprobable.

Take English text where our symbols could be letters of the alphabet. These are not equiprobable, if you select a letter from a book at random you get a different distribution than 1/26. If you took as your symbols the individual bits that would encode those characters in ascii you would get something closer to equal probable symbols. Another choice for symbols would be the words in the book.