The *total* light, integrated across the area of the FF sensor at f/8, is indeed greater than the crop sensor at f/5.6, in terms of exposure, f/8 is still f/8. If you shoot the same scene at the same time and the same ISO with both cameras set to f/8, they will both require the same shutter speed.
What you're saying is equivalent to saying "The sun is brighter at my neighbor's house because he has a bigger yard."
Yes... and no. If all you care about is getting equivalent brightness and you ignore all other aspects of the image then sure, f/8 is f/8 is f/8.
(Strap yourselves in, kids, class is in session.)
But 1) not all cameras produce the same brightness at the same ISO setting, and 2) smaller sensors produce more noise at the same ISO (assuming the cameras are of a similar age and resolution), so you're not ending up with equal quality.
Even if we ignore #1 for now, equalising #2 requires you to get more light onto the smaller sensor so a lower ISO can be used. (Or less light onto the larger sensor, to use a higher ISO.) We do this by changing the aperture, since changing the shutter is usually more destructive and causes other differences in the resulting images, while changing the aperture also allows us to equalise the depth of field at the same time. The difference in both depth of field and noise between 35mm and APS-C tends to be
roughly one stop and a quarter, so it's an easy compensation to make by just changing the aperture by one stop. (And for 4/3rds cameras to 35mm, it's two stops.)
In this sense, f/8 is no longer f/8 when moving to a different sensor. 1/100 f/8 ISO 800 on a 35mm sensor and 1/100 f/8 ISO 800 on an APS-C sensor may give you the same brightness and motion blur/freezing, but that APS-C shot will be noiser and have a deeper depth of field. If we change the APS-C camera to 1/100 f/5.6 ISO 400, now we've got the same brightness, (roughly) equivalent depth of field, and (roughly) equivalent noise.
So in a
total equivalent sense, f/8 is
not f/8 when you're talking about different sensor formats. Aperture
effectively changes just as much as focal length does. ("f/8", after all, does literally mean 'focal length divided by 8', and if you paid attention in algebra class you know that you can't change 'f' without the rest of the maths changing, too; 'equivalents' and 'crop factor'
are applied to the whole equation, not just the focal length.)
To use your garden analogy, yes, having a bigger garden does not mean the sun
itself is brighter, but it
does mean that garden recieved more sunlight. On average throughout the year, in one minute and per square meter, the amount of sunlight that hits the ground is 6kWh/m². If someone's garden is four square meters then in one minute their garden recieved (or saw, caught, or experienced, or whatever other term you care to use) 24kWh; if their neighbour's garden is eight square meters then that garden, in the same time, recieved 48kWh of sunlight. It's double the area so double the sunlight landed on it, even though the sun itself was putting out light at the same rate.
A garden or a sensor, it's the same thing. It's not that the light is brighter, but that that more of the light is able to be captured. The smaller sensor captures less of the light in the first place (and is less light-efficient at the same resolution), so the signal has to be boosted more and hence you end up with more noise. Open up the aperture, let more light in, less boosting (ISO) is needed, and now everything is nice and even.
But then there's point #1 earlier, which causes big problems when comparing different brands, since every manufacturer rates ISOs differently, and in particular Canon and Fuji cameras are sometimes actually shooting at as little as half the stated sensitivity. So, in some ways, the biggest problem with getting equal exposure between cameras isn't the aperture, but the fact that ISO 100 simply is not always ISO 100.