Quick release and tires moving in the frame?
The clamping force should be what keeps the axle in place, not the drop out. As you found out hub brakes, such as disc brakes, put more stress on quick release skewers than  rim brakes. While drop out design can help (e.g., forward facing), this ultimately is a failure of the skewer clamping force. 

Unfortunately, not all skewers are created equal and many bike builds use cheap skewers.  In fact, most quick release skewers I have used are deficient in this regard allowing the front wheel axel to deflect under heavy braking. This is one of the reasons thru-axles have  become so popular for disc brake applications.

That said, quick release can still work if you get the right skewer:


High quality 
Internal cam design 


Shimano skewers (Deore and above) tend to be some of the best in terms of clamping force. See if you can track one down for the front, a better skewer for the rear isn’t required as the braking forces are different.

Note that the bike was spec’d from the factory with a quick release.