I did not see where this using either does or doesn't support ASIO drivers. All I could find on this unit is that it's discontinued and no mention anywhere I looked about the drivers it uses.
It's not a case of does MC7 support ASIO... it's rather a case of does the interface support full native ASIO drivers or does it use codecs and wrappers as well as MME and other drivers instead. MC7 will list the drivers is sees in it's input and output options and it will try to use them. Whether they work well with the soundcard is a totally different issue.
The other programs you mentioned will tend to work well on MME drivers on the laptop factory card as you mentioned, so you really can't compare them in this manner. MC7 will not run smoothly on the factory card with MME.
Most of the better interfaces will run the Native ASIO drivers. If your interface doesn't, you might want to consider selling it on eBay and buy one that does. Check your users manual because that should list the driver modes it supports.