If/when private sector operators come forward with proposals, I will definitely pay attention. But the track record of public-private partnerships like this suggests that the private operator will build little-to-no additional infrastructure, and instead just look for ways to use existing infrastructure. In the O'Hare travel market, this means either a sub-optimal train trip via Metra or a trip via the Blue Line that places serious delays on normal L trains and those who ride them.
IF Metra were already electrified, then tunneling beneath the airport and building a new rail terminal there would probably be less than $300 million (the
Hiawatha airport tunnel is a decent yardstick, with similar soil types and length). However, Metra isn't electrified, so you can't run its diesel trains into a tunnel. And even if you purchase dual-mode locomotives, you still need to run FRA-compliant equipment, and bore a tunnel big enough to squeeze them into.
Building passing sidings on the Blue Line isn't a great idea either. The Blue Line has to shut down fairly often for suicides, accidents, bomb threats, and occasionally derailments. Now into this, let's throw a system of passing sidings that shoots to hell the idea of frequent, regular service for the Blue Line. Just as a good experience with an O'Hare Express train could very well bring in new passengers, a bad experience - say, getting caught in the Milwaukee-Dearborn subway - could turn off these global-city visitors and make them vow to never visit Chicago again.
I'm not saying it's impossible to set up a good service, but it's not the kind of turnkey operation you want to hand over to a private company. It will require substantial and costly new construction. If and when there's surplus money lying around to build a tunnel that brings Metra into the O'Hare terminals area, with electrification and all, then we can set up a good service.