Major reason is that the DSLR lenses are optimized for Phase Detection. Every component of the lens is tailored towards quick movement and stopping the glass in precisely picked moment. Contrast detection on the other hand works best with stepper motors capable of quickly switching directions so that you can move lenses inside back and forth looking for highest contrast on the image.

Phase detection knows straight away where the focus is and how much should the lenses move to achieve perfect focus. Contrast detection needs to "find it". This forces different engineering solutions in lenses manufacturing.

Also the DSLRs are usually made with live view as an afterthought. Most of the manufacturers think of it as an manual focus assist. They don't try to create fast AF for video as they know that pro videophotographers usually relay on manual focusing (not that they have any choice with such performance...) while photographers usually use viewfinders anyway. Hence they rarely have dedicated processors for contrast detection, and if the main CPU is occupied with focusing - it doesn't perform as well as dedicated unit.

Also the statement in your question isn't entirely true. Sony's SLT DSLRs have AF that's much faster in live view than Mirrorless, as SLT design basically allows camera to fully utilize it's PDAF (Phase Detection Auto Focus) sensors all the time during live view. So you get DSLR-quality AF with Live view at the same time. Also the older generation of Sony DSLRs offered Quick AF Live View - which never flipped the main mirror up for Live View - instead it used secondary sensor in viewfinder allowing DSLR-quality AF for a price of additional delay before shooting photo (mirror had to flip up in order to capture the photograph).