Do not tow with a class 1. If you must, tow under 45mph and just like a few miles at most.
Come on now, it's not that bad. Mazda states in the literature that comes with factory hitch a maximum trailer weight of 1,500 lbs and maximum tongue weight of 200 lbs., and also that it is designed to safely tow up to the limits stated in the owners manual. The owners manual states the CX5's maximum TTW is 2,000 lbs. TTW is the total load including trailer, hitch, passengers, and cargo. In those ratings the owners manual makes no distinction between classes of hitch, so even if you managed to bolt a fifth wheel saddle back there it would not change Mazda's TTW recommendation. Owner's manual and hitch literature make no mention about speed limits when towing.
The factory hitch attaches directly to the frame, to points that were engineered for that specific purpose. It weighs 53 pounds; I think Mazda rating of it as Class 1 is conservative; I suspect it is just as heavily built as the Curt class 2.
In practice I would not hesitate to pull a 1500 lb trailer with a tongue weight up to 150 lbs; above those I would be concerned about the vehicle's overall ability to handle the weight--brakes, frame, suspension, drive train, wheelbase--more than the ability of the hitch to handle it.
You might have guessed that I have the factory hitch with no regrets. I like that it tucks up as high as possible, improving rear ground clearance over aftermarket receivers.
With that said, I would be less confident in any 1.25" hitch's ability to handle carriers. If you want to put a bike carrier back there or hang a big box off of it, then a 2" receiver makes more sense. Personally those accessories bother me, seeing them hanging off the back of vehicles looks 'wrong' to me, seems like an awfully big lever to put on any drawbar that was engineered for straight-line load bearing. But that's just me. Interestingly, Curt recommends a stabilizing strap for their bike carriers, which attaches the top of the carrier to a point near the roof such as the top of the hatch. Those make sense to me. With one of those I would feel ok about hanging bikes off a 1.25" receiver.
Cheers