The funny thing is that it actually used to do just that, in addition to having a Westchester branch running N-S to Mannheim and 22nd:
Source: www.thetrolleydodger.com
Sadly, the line was ripped up in 1951, right before the area experienced the post WW2 suburban boom. Had it stayed on for a few more years, ridership would have justified its existence and the area may have developed in a much more urban manner, at least near the train stops.
You can still see the old ROW with the property lot lines and housing development:
https://www.google.com/maps/@41.8558...m1!1e3!5m1!1e2
Look for the curve that is just northwest of the Mannheim & Cermak intersection.
I think extending the blue to Mannheim could serve a purpose, offering a route downtown in between the UP-North and BNSF Metra lines. The area is fairly dense and built up, so ridership numbers would be plausibly good. Having stations out that far from the Loop however would definitely necessitate the need for express trains, if not all the way from Mannheim then at least from Austin to Clinton. Otherwise it would be a 90 minute train commute, which would make driving on 290 fairly comparable, if not faster.