Well, the Waytorussia folks tell me the rate on their website is a typo -- a taxi from DME to the city is $53. At that price, it's an offer I can refuse.
The Aeroexpress now has a better mid-day time table now that the literature suggests. If you're coming in on the AA flight from Chicago, there's apparently a train service every half-hour.
http://www.domodedovo.ru/en/main/getting/1/aero/
I understand you need to be a little careful following the signs at DME to the train, but I presume it's managable. I've riden subways from Buenos Aires to Beijing, so how hard could Moscow's be?