My own experience basically agrees: I’ve read and enjoy Skiena, it’s written in clear style and it’s the “cover to cover” text for a working developer or for interview practice. But I also have TAOCP and CLRS on my shelf, and I haven’t read either of them. I’ve certainly used both of them a lot, but I simply haven’t read all of them because I use them more as reference texts.
Personally I find it bothersome that these textbooks are written with idiosyncratic pseudocode. In my opinion, many of these authors lose their grasp of common implementation difficulties if they don’t provide students with working code that will compile and run. It’s easy to throw down the theory and some okay-ish pseudocode while in effect saying, “...and the rest is just a minor implementation detail, which I leave as an exercise to the reader...”