They are not poorly formatting VCF files. They use an extended, non-standard format (i.e., non-standard != poor). For example standard GATK format is not amenable to triallelic sites; they had sites that are biallelic in modern humans, but the Neanderthal/Denisovan was heterozygous with a third allele. (http://www.sciencemag.org/content/suppl/2012/08/29/science.1224344.DC1/Meyer.SM.pdf pp. 16-20; http://www.nature.com/nature/journal/v505/n7481/extref/nature12886-s1.pdf p. 14). I've contacted one of the creators of the files in the past, and he has said that trying to use GATK/vcftools with these files would not be a good idea and that python or pysam would be the best way to go.
You are jumping to a lot of conclusions about the filtering. Based on the Meyer link above, they used more than what you think with multiple iterations of genotyping. (These files are from large scale ancient DNA genome projects, they are not going to be that sloppy).
Given the fact that the VCFs are from ancient DNA, GQ is probably important (and there are some analyses in those supplemental docs indicating that lower GQ values have biases some dating analyses)
can or cannot use GATK/vcftools?