500lbs and 18" wheels is not trivial with an inclination to climb
Hmm I hoped my explanation was easy enough to follow,
especially since your values are very close to my examples (except for the weight!)
5° inclination, 9m/s desired top speed.. is identical...
Anyways.. let's check...
500 lbs = 227 kg (rounded)
so to get up such 5° inclination:
227 kg * 9.81 m/s² * sin(5) = 194 N
add speed (times 9 m/s):
1746,76 Watts
to go up that inclination at 20 mph...
but that's not what you asked for.. you asked for 10-15 mph uphill..
10 mph = 4.47 m/s ( or 868 Watts )
15 mph = 6.7 m/s ( or 1300 Watts )
as you can see.. uphill is not an issue with just one 1kW my1020 if you're okay with just 10mph.
flat road top speed of 20 mph is doable with again just over 800W (820 or so for a 227kg kart)
no worries there.
Again, a single motor is enough if you do not insist on 15mph or more up a 5° inclination.
And, with a 7:1 gear ratio (well frankly.. IDK the exact rpms under that kind of load.. but somewhere between 6.9:1 and 7.4:1 is very very likely)
should dial in for the 20mph top speed as I said (the numbers above turned out to be correct

376 wheel rpms needed with 18" wheels)
And I'm rather confident that the motor is capable of doing exactly that rather easily to be honest.
Anyways, a second motor will make it... uhm let's say .. snappier

and overall make it appear effortless. (even uphill at speed)
you could even add a few mph to the top speed (i.e. raise the gear ratio to say 6:1 for ~25mph)
Anyways... 227 kg is NOT much considering the weight of an adult driver (say 75kg) and a set of batteries that last longer than 30 minutes (another 75kg)
a set of wheels (say 20kg) leaves no more than 57kg for chassis, steering components, axle, sprockets, brake, seat, motor and controller and whatnot...
So I wouldn't gear higher than 6:1 really not with tyres that large and the good chance of unwillingly adding another 50-60 lbs
[
EDIT]
Why do moth motors need to see the exact same load?
Is that statement theoretical or have you tried this in the real world?
.......
How does one find matching "Chinese" stuff?
Even if a retailer just got a fresh order in, how do you know what matches what or what came from what batch?
Most sellers don't know shirt about the stuff they sell. (especially Chinese stuff)
real world (but smaller "non-EV" motors I admit)
The thing is, that even IF both motors are running at the same speed
(both power one axle/wheel whatever)
they do NOT do the same work, a 5% difference in current results in a 10% difference in work (one doing 55% of the work the other just 45%)
and while that's not an issue as long as the motor that's doing MORE work is below it's limits.
think of it as running a 800W and a 350W motor together for a 1100W task.
(I bet you cringe at that idea)
it's exaggerated, true, but the same rule applies.
it starts to become a huge issue as soon as one motor overheats,
since then, all of a sudden it looses a serious fraction of it's power (likely fails completely) and the other motor suddenly has to do all the work and dies soon after.
We (well most of us here) tend to reach for the limits (more speed, steeper hills to climb, testing the equipment as hard and relentlessly as possible)
That's why a well levelled set is the only combination I would suggest.
Reliability is essential if you're not willing to spend that money twice.
And you're right, you cannot be 100% sure about matching motors and or controllers.
but if a seller got a fresh batch directly from the mfg and not some wholesale chances are it's one batch.
(do you think they have spare motors on layaway to put in a box six weeks later or something? I doubt it.. from the belt to the box as they are tested...)
same holds true for the controller.
If you do not buy off ebay but from alibaba you can contact the manufacturer they will even send you motors with consecutive serial numbers if you ask them to I bet
And while there's still a chance the motors do not perfectly match,
it's the best chance you got.
'sid