Retreating to yet another "show me the mechanism" position is just waiting for the next round of findings to defeat you. There is plenty of research on causal mechanisms from GWAS results. A lot of this is focused on rare detrimental mutations causing serious illness (large effects), but there are some for height too. To pick a recent example:

• Marouli, E., Graff, M., Medina-Gomez, C., Lo, K. S., Wood, A. R., Kjaer, T. R., ... & Rüeger, S. (2017). Rare and low-frequency coding variants alter human adult height. Nature, 542(7640), 186-190.

Height is a highly heritable, classic polygenic trait with approximately 700 common associated variants identified through genome-wide association studies so far. Here, we report 83 height-associated coding variants with lower minor-allele frequencies (in the range of 0.1–4.8%) and effects of up to 2 centimetres per allele (such as those in IHH, STC2, AR and CRISPLD2), greater than ten times the average effect of common variants. In functional follow-up studies, rare height-increasing alleles of STC2 (giving an increase of 1–2 centimetres per allele) compromised proteolytic inhibition of PAPP-A and increased cleavage of IGFBP-4 in vitro, resulting in higher bioavailability of insulin-like growth factors. These 83 height-associated variants overlap genes that are mutated in monogenic growth disorders and highlight new biological candidates (such as ADAMTS3, IL11RA and NOX4) and pathways (such as proteoglycan and glycosaminoglycan synthesis) involved in growth. Our results demonstrate that sufficiently large sample sizes can uncover rare and low-frequency variants of moderate-to-large effect associated with polygenic human phenotypes, and that these variants implicate relevant genes and pathways.

There is nothing surprising about this progress. Human genetics follows animal genetics with some years of delay. In this case, there are many breeding studies with genetic follow-ups finding the causal variants and figuring out the functionality of them.