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If there is no OX signal the ECU obviously can't go into closed loop operation.... Thus you're stuck in open loop operation full time. If that's wire is broken it will absolutely cause the vehicle to run rich (to some degree all the time and under all circumstances). There may be other problems but a broken OX signal wire must be addressed before you bother with anything else.

The CEL only turns on for certain high level codes and only remains on for persistent or severe diagnostic events. There are very few which trigger the CEL permanently. The supposition is that a vehicle's owner will take their vehicle to the dealership for regular maintenance where it will be checked for codes on a regular basis without alarming the owner if a minor problem is recorded. This is still the case with current vehicles although there are a lot more possible codes and more of them will present at least a temporary CEL. It's a good idea to check for codes whenever you change your oil (or more frequently if you like).