Wanted to share my 2 cents...
1. Dealer website awareness / Branding is necessary for any dealer. This is true regardless of your intent to post to 3rd party classifieds sites. The bigger your dealership / group, the more you need to spend on branding and direct traffic to your site.
2. Regardless of you size, you will need a 3rd party listing site. No matter how strong your brand (unless you are the ONLY dealer in town), an increasingly large percentage of car shoppers will want to take a look the general market outlook. They want to see what's out there. And quite a few new car purchases are impulse purchases, and these people want to know if their specific model requirements are available (RIGHT NOW that is). These people will want to cast a wide net. So eliminating 3rd party listing sites may work, and can even continue to keep you profitable in the short term. But you will be leaving some business on the table.
3. #2 is even more true for used cars. Very few people in this day and age shop for used cars by dealer lot. They shop for "Used Chevy Malibu with less than 60,000 miles", and not "let me check out the used inventory at dealer XYZ".
4. Absolutely agree with Joe Pistell above. Car shoppers will be "ROBO" at least into the foreseeable future. Would you buy a house using a shopping cart on Amazon? Ebay perhaps? A car is no different, and in many cases a lot more emotional than a house.
5. As for the question, "who's next". Watch out! I am working on it...