Approximately Optimal Mechanism Design vs. Revelation Principle
A key part of the paper is that they are interested in 


  computationally efficient truthful mechanisms


The VCG mechanism is a truthful and efficient direct mechanism, but it is not computationally efficient. In many combinatorial auctions, it is impossible to compute the VCG transfers in a reasonable amount of time using current technology.

In the past, economists have largely ignored the problem of computational efficiency in mechanism design. This problem has been mainly addressed by computer scientists. Read for instance the introduction to Milgrom and Segal (2020) Clock Auctions and Radio Spectrum Reallocation. JPE