Having spent a terrible summer working at Target as a cart attendant, I can explain at least part of the reason why their restrooms are gross. The job of cleaning the restrooms, as well as every other awful job that nobody wants to do, is given to "cart attendants", who applied thinking they'll just be pushing carts around all day and don't find out about these "side tasks" until their first day on the job. So these side tasks often go neglected out of spite and a lack of willingness to deal with that crap (figuratively and literally) for just above minimum wage. Why a gigantic store like a Super Target wouldn't hire a separate person as a "custodian" or "janitor" for such things is beyond me.
My experience with Target restrooms has been mostly positive. They also tend to be easily visible, near the entrance and signed, unlike many other big box stores where you have to ask someone because its impossible to find the restroom.