Indeed, the integrand is basically zero for the majority of the interval. I think QuadGK samples quadrature nodes adaptively and misses the region where the integrand is non-zero. I also run into this from time to time when numerically computing CDFs by integrating PDFs. I think I solved this by adding a few additional quadrature nodes around the mode:
quadgk(x->.., -50, MODE, Inf)
This usually works fine.