^^^ That's a great point, not all cars can tow or should tow.
In this case, would be a non-issue on the Corolla, from a legal standpoint, as the is spec'd to take a Class-I trailer with a max towable limit of 1500lbs.
With his expected trailer weight that will stay inside of the car's max GVWR. Granted, this trailer rating system is pretty poorly explained, even the sale's people don't know - very consuming concept.
Example (rounded the numbers to make the math simple):
9th gen Corolla GVWR is about 4100 lbs
9th gen Corolla curb weight is about 2600 lbs
4100 lbs - 2600 lbs = 1500 lbs, which just happens to be the max "tow rating".
ACTUAL towable weight is NOT 1500 lbs
1500 lbs - [ passengers + cargo + fuel + (say 850 lbs) + trailer tongue weight (600lb trailer, 15% is 90 lbs) ] = 560 lbs, well inside the GVWR
The remaining capacity, I try and keep the trailer weight below that. That trailer can technically be higher, as you just need to account for the tongue load - but I find that if I add the weight of the trailer as well - the car's brakes and engine can keep up pretty well.
Steep hills and long grades, especially if you have the A/C running - that will take a toll on any car. Most find that their max tolerable towing speed is around 50-55MPH anyways. Depends on your route and if it is windy that day or not.