An f/2.8 on APS-C allows the same.amount of light as f/4.0 on FF and has the same depth of focus.
The 2 are directly linked.
However,the problem is that if you are shooting at F2.8 that is more or less wide open for many zooms,whether they are APS-C or 35MM frame. In the end most zooms are much softer wide open anyway,so you will need to stop down slightly even on APS-C,and is what I do for both my APS-C and FX systems,and I always tend to shoot more in what is the optimal sharpness point for my lenses,which is between F5.6 to F8.
Also if you in very low light conditions,and have to shoot wide open,forget DoF, you are mostly likely using the lens at its softest aperture and focusing becomes probably less critical in low light,plus OFC the effects of noise reduction,and it doesn't matter what format sensor you will be using for a lot of general purpose pictures.
Plus people really need to get it out of their head - the 35MM frame is not "BIG". It isn't - medium format is what was considered the start of bigger formats(medium not big). You only have to look at some 6X4.5 film to see that:
https://www.flickr.com/photos/bwrightphoto/5388575653/
The 35MM format came along since cameras were truly huge in the past,and IIRC was derived from people modifying cine film(IIRC) to fit in cameras.
Plus the issue,a lot of sensors are designed with high MP counts which means the surface area of the photo-detectors is generally smaller than some of the more common 35MM frame sensors,and this is why some of those 35MM frame sensors have such good DR and low light capabilities at a similar technical level. But then as seen with some of the very high MP 35MM frame sensors,DR and low light sensitivity suffers.
That is an issue in itself - technology can help in a way(microlenses,re-arranging the processing circuitry and certain back end processing,etc) but people concentrate on the physical size of the chip,which is not really the most important aspect of it. If you look in science,for example,when people are looking at imaging cameras,they rarely go by sensor size,but photodetector area,readout speed and other more important metrics.