No, it wouldn’t (of course ignoring the fact that the equivalent to saying ‘is reading’ would be 読んでいる not 読む). That is not what anyone saying or writing that sentence would possibly mean. By this logic, the sentence:
must mean ‘Today reads’ which is just illogical. I can even say:
Which again does not mean ‘today reads a book’. It would be something along the line of ‘As for today, I will read a book’ with the nuance that maybe I don’t read a book every day or I didn’t read one yesterday but I will today.
Also, は introduces a topic it does not ‘show the subject’ (not sure what that means honestly).