EV is a measure of illuminance, which is defined in the link you provided as "luminous flux incident on a surface, per unit area". You are correct in stating that when if you keep field of view, depth of field and subject brightness constant: $$\mathrm{EV_{crop}} = \mathrm{EV_{ff}} \times c^2$$

however since: $$\mathrm{Area_{crop}} = {\mathrm{Area_{ff}} \over c^2} $$ and $$\mathrm{Light_{total}} = \mathrm{EV}\times\mathrm{Area} $$ we arrive at $$\mathrm{Light_{crop}} = \mathrm{Light_{ff}} $$

In other words your APS-C system will collect more light per unit area of the sensor, however by virtue of a larger sensor a FF system will collect the same amount of light in total.

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However, when comparing systems in any practical sense you have to take lens availability into account. For a given full frame lens there may not exist a lens for APS-C with focal length c times shorter and f-number c times lower.

From 135mm and up you can generally achieve equality in light gathering, let c = 1.6:

135mm f/2.0 -> 135/1.6 = 84.3, 2.0/1.6 = 1.25 -> 85mm f/1.2
500mm f/4.5 -> 500/1.6 = 312.5, 4.5/1.6 = 2.8 -> 300mm f/2.8

In the normal to short tele range the best you can hope for is to maintain the same f-stop, which means projecting the same amount per unit area onto the sensor, meaning the larger sensor gathers more light total.

     FF         APS-C
85mm f/1.2 -> 50mm f/1.2
50mm f/1.4 -> 30mm f/1.4

At the wide end lenses for full frame can be significantly faster, giving the full frame system more light per unit area and more area for a significantly greater light gathering ability:

     FF         APS-C
24mm f/1.4 -> 14mm f/2.8