Actually we do have a lot of Neolithic mtDNA samples, and the percentage of haplogroup H gradually rises from the Early to Late Neolithic to nearly 35%. We also have over 100
Bronze Age Steppe samples and mtDNA H makes only 25% of the total (and only 10% in Andronovo).
I believe that the main reason that hg H appears so low in Neolithic samples tested so far is because H was more common among Mesolithic southern Europeans, and especially in Iberia. Most of the Neolithic samples were tested in Central Europe, where Mesolithic inhabitants belonged predominantly to U2e, U4 and U5. IMO, only a few H subclades came from the Middle East with Neolithic farmers (e.g. H5a, H20). The rest, including H1 and H3, were probably native to Mediterranean Europe.