I can vouch. My parents used to own an Omni that worked fine for about six months.
I'm guessing the two posters' parents' Dodge Omnis were simply lemons off the line. (I read in one consumer magazine once a letter from an owner who had purchased a car based on good reviews and a good reliability track record, but his car was a piece of junk instead. He took it to a mechanic who vouched for the issues the car was having and offered to fix it but it would void the owner's warranty. Other car owners never had the same problem, so this was an apparent isolated case.)
While Consumer Reports magazine had a negative review of the 1978 model branding it "Not Acceptable" due to serious defects the car did improve by the early 1980s, as far as what I've read in consumer satisfaction and reliability reports.
And no, I never owned an Omni/Horizon.
Brian