In theory that looks convincing but on a practical plan, depending on what you are photographing, it doesn't always work like this.
I'm a pro concert photographer shooting mainly classic concerts so you can understand that low light is my daily bread. When I shoot a full orchestra on stage, my main concerns are shutter speed high enough to avoid motion blur, and DOF deep enough to get the whole orchestra in focus.
I shoot mainly M4/3 so let's assume that in an average concert hall lighting I can shoot at f/2.8 and still have enough DOF to get everyone in focus, and ISO 800 would get me the shutter speed needed to eliminate motion blur.
If I would have used a FF camera instead, for the same DOF I'd have to shoot at f/5.6 and then in order to maintain my shutter speed I'll have to boost up ISO to 3200. And this is how the FF two stops theoretical noise advantage flies out of the window.
of course, this is a specific case but I mentioned it because in real world photography, things do not always behave as they are in theory.
Moti
The issue is that the "full frame advantage" isn't what most people think.
Assuming your shutter speed is constrained, similar sensor technology, and the same angle of view; low light performance is generally a function of depth of field. This is independent of sensor size.
As you have noticed, at the same angle of view and same shutter speed, your full frame camera needs a 2 stop higher ISO in order to get the same image as a 2X crop body.
The real advantage of larger sensors, is that for a given angle of view, you generally have the
option of a larger aperture diameter. That larger aperture diameter gives you both better low light performance, and shallower depth of field.
If you are in a position where you can tolerate that shallower depth of field, the larger sensor gives you that option. In a situation like yours, where you can't tolerate shallower depth of field, there isn't an advantage to moving to a larger sensor.
This isn't to say that full frame is "better". it simply means that full frame offers more options. If you need/want those options, then there is an advantage to the larger sensor. If you don't need the option, than there's no advantage to a larger sensor.
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For instance, I have a 50mm f/1.8 lens for my full frame camera. This is a common and inexpensive lens. Wide open it yields a shallow depth of field and reasonable low light performance.
On a 2X crop body, I would a 25mm f/0.9 lens in order to match that shallow depth of field and associated low light performance. That's not nearly as common a lens.
My local Best Buy has in stock a 50mm f/1.4 or a 50mm f/1.2 lens for my camera. A 2X crop body would need a 25mm f/0.7 or f/0.6 in order to match that depth of field.
If you seldom shoot 50mm wider than f/5.6, then you can shoot your 2X crop with a 25mm at f/2.8. If that's your shooting style, you won't benefit from a full frame.
Unless you need/want to shoot with shallower depth of field, there isn't a significant advantage to larger sensors.