Your claim that "there is a lack of genetic diversity" in mitochondria is not correct. There is no connection between recombination and mutation rates per se. In fact, mitochondria have much higher mutation rates than nucleus and different types of mitochondrial DNA can even co-exist in one organism, a phenomenon called heteroplasmy (which is found at a frequency of ~90% in human).
But: there is a connection between mutation fixation rate and effective population size. Because mitochondria are normally inherited through maternal line, effective population size for their genome is smaller than that of the nuclear genome, and thus any mutations in the mitochondrial DNA reach fixation faster, but again this is counteracted by high mutation rates.
There is no problem with functionality: as always if something gets broken (and it is easy to break something in the mitochondrial genome, since genes are normally tightly packed in it) respective genotype is eliminated or reduced in frequency by natural selection. In the case of mitochondria this selection acts both upon the level of individual organisms (intra-individual somatic pool of mitochondria: positive and negative selection) and the level of populations (mostly purifying selection in humans). It is nevertheless true, that given the virtual lack of recombination, selection becomes the only force of purging deleterious mutations.