Yeh, Mongols relied on speed a lot. Not just on horseback, moving around fast, but also in terms of firing rate. You can't fire rapidly with a crossbow, so you'll either have riders firing all at once, in volleys, or sporadically, with shots very spaced out and random. With bows though, you can have a continuous stream of arrows raining down on the enemy because every rider can and will be firing arrows back-to-back, so there is no pause. Cranking up a crossbow on horseback, especially on the move, has to be a pain in the ass too. I can't say crossbows would have been my first choice.

EDIT: Just realized the above post had nothing to do with speed. Sleep-deprived again, don't mind me. Still a good thing to remember though.