I can't agree more with your judgement. I've read all these three books and the fact is just like what you mentioned. CLRS and TAoCP use a lot of space to prove the algorithms correct(especially TAoCP, it even uses different ways to prove one algorithm) and that makes me feel dizzy. Perhaps I'm just a programmer not a computer scientist.
But I have to say that use some time to read CLRS and TAoCP is worthwhile. It will help u know more about algorithms and avoid the condition that just know how but not know why.
The author forgot to mention how to cultivate an interest in coding which, I think, is a very important part.
I have seen so many CS students that didn't like programming at all. They just finish the tasks their teachers give them and never want to see one more LoC in their idle time(Ironically, they still want to be great coders when asked).
In my opinion, the first step to level up to a programmer is to "force" oneself to love coding.
But I have to say that use some time to read CLRS and TAoCP is worthwhile. It will help u know more about algorithms and avoid the condition that just know how but not know why.