It actually does allow for variation, but at a much slower rate. Asexual offspring will still experience transcription errors and other random mutations. Naturally this is less of an issue for organisms that reproduce at a high rate than it would be for complex organisms.

Many species reproduce asexually - many plants, for example, or bacteria (although they can share DNA plasmids directly). Evolution happens just fine, as variation is supplied by mutation. Nevertheless, sexual reproduction is the norm.

In terms of selection, it is sexual reproduction that is a massive disadvantage. From the gene's eye point of view, sex involves a "two-fold cost". Essentially any given gene has only a 50% chance of being in an offspring, compared with 100% for asexual reproduction.

Something needs to overcome this two-fold cost, most probably the fact the sexual recombination allows discovery of fitter genotypes for disease resistance and host/parasite coevolution.

https://en.wikipedia.org/wiki/Evolution_of_sexual_reproduction