The rate of evolution in an asexually reproducing species will always be significantly slower than a species that reproduces sexually, and over time it will be out-competed

Again: not necessarily. That there are still asexually-producing organisms suggests that it's not enough of a disadvantage for the environment they're in.

This form of reproduction can also be an advantage: the rate of reproduction tends to be faster and more offspring are produced. In the case of this organism, instead of one queen producing all the offspring, every ant is able to produce offspring.

tl;dr: it isn't the case that sexual reproduction is always favoured over asexual reproduction. Evolutionary pressure isn't fixed like that.