Emirates has a 7 kg (15 pound) limit for carry-on luggage. ... But my carry-on bag, a 21-inch bag with a hard shell, weighs about 5-6 kg when it is empty. ... I can't see how they could expect me to fly 20+ hours (total, one-way) with only 1-2 kg of actual "stuff"...
I just want to point out that your line of thinking here is incorrect. You are framing this as Emirates' fault. However, Emirates does not decide the weight of your carry-on bag; you do. Your 5-6kg "bag" (more of an armored case), which consumes virtually all of your allotment, is your choice, and it is obviously a very, very bad choice. To call this Emirates' fault is absurd; your bag choice just doesn't work, and for some reason you can't see that.
I get where you might be accustomed to doing this on other airlines, but then, I would say fly those other airlines. Clearly Emirates plays by different rules, and you need to follow them to collect the cost savings that drew you to Emirates in the first place.
Obviously, the weight alone is a showstopper. Nope. Nope.
Certainly, at 21", there is no possibility of putting this under your seat, and it will have to go in an overhead bin. So claiming you need it for things needed during your 20+ hour flight doesn't make much sense either. You would not be able to access this bag without making a huge fuss: pulling it down out of the overhead (which might be quite far from your seat!!!) and jostling other people's bags as needed, taking it to your seat, opening it only partially because there's no room to open it fully, then rifling around in it to get the item of your interest, then taking it back to the bin and again rearranging everyone else's bags so your bag fits. What a fiasco. Maybe you're used to doing that whenever you fly, and don't realize how absurd it is, but try it.
The 7kg maximum weight greatly limits the potential volume of the items. You don't need, and can't use, anything near a 21" bag.
In fact, 7kg of stuff would easily fit under your seat. And now you're talking! This is a space that is yours, you control it, nobody else can damage a bag there, and you can access it almost anytime during the flight with very little fuss.
Further, you don't even need rollers for 7kg, so there's no reason for a heavy hardback, extendable handle, rollers, or any of that deadweight. All you need is a shoulder strap that might weigh 0.1 kg.
There's a parable of the person trying to push a river. They mean when someone gets up a full head of steam, hell-bound and determined, to do something at all costs that is in fact totally impracticable. In this case, you are "pushing a river" with your willfulness to use this very poor choice of luggage. As such, our job is to snap you out of that, and bring you back to practicality.
You need a new bag :)
I recommend a very light 0.5kg or less canvas bag with a shoulder strap. No wheels, no handle, no rigid structure of any kind. It needs to be a feather. And it needs to go under the seat. Its job isn't to protect anything, that's the job of you and the camera bag.
If the electronics going in this canvas bag need better protection than the item case (e.g. camera bag) provides, then upgrade the item case. Because armoring the camera bag or using an Otterbox for a phone is much lighter weight than armoring all your stuff for the sake of the camera.