Unlike a lot of things in HVAC, the location of return air grills is definitive.
If you can not remove the air to be treated from the location and process it in an HVAC machine, the system will not work.
Hot air rises and cold air falls. When you have a two story house with one air-conditioning system and there is not a return on both floors, or in this case are both located on the same floor, it simply will not work. The duct system can only be designed for heating or cooling. It will not work efficiently for both. Sometimes we can “get by” with marginal applications, however the case in question is not a marginal case.
The only way one HVAC system can adequately perform in a two-story house is if a zone control system is installed and there is a return register on both floors. If you do not have an automatic controlled zone system, then seasonal campers must be installed that are manually change between winter heating and summer cooling mode.
It is seldom considered, however balancing the return air is as critical to performance as balancing the supply air.
Air is a fluid. It contains a considerable amount of moisture which does not like to change temperature readily. Blowing cold air into a hot room does not change the ambient temperature of the room efficiently. You cannot remove the moisture from the air by just blowing cold air into it because you must move the air through the HVAC system to remove it.
Heating systems are generally less critical if a return is located on the lowest floor as cool air is removed and hot air rises to the second-floor (just leave the doors open). So it is less noticeable in the winter than it is in the summer. In southern locations cooling is the most important function of the HVAC system, so the system should be designed cooling. The heating system seems to get by without any complications in most cases.