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Sensor size matters, but not in these insignificant steps from 1/2.8" to 1/2.5" or from 1" to M43 for example. Sensor technology and SNR are normally more important than going from a smartphone to a 1" point and shoot. I will argue there are two things looked over things that make sensor size important though:
>Pixel density
Smaller sensors will have higher pixel density and lower aperture size, which means less light and more pixels to fill with light = a rough time for anything but good lighting conditions. However, this means you can get the same "zoom" with a smaller lens. This is why smartphone lenses are tiny but LARP as 24/28mm or even 50mm lenses. Extrapolate that to more aggressive focal lengths like 400mm and suddenly it's way cheaper and lighter and smaller to get a decent photo with APS-C versus a Full Frame kit.
>Lens pupil entrance size (aka aperture size not f/stop)
Bigger apertures let in more light. When photography is literally the capture of light, that makes aperture important. Now the fucked thing is, marketing for anything but full frame systems will cheat and slap "f/2.5" or something on their lenses or phones because obviously the lower the ratio the better! True, but overall diameter is still what actually matters.
An f/2 phone lens is about 2mm of aperture diameter. An f/2 FF lens is 25mm. But it's sexier to slap f/2 on the marketing page than "2mm wide aperture".
Anyway, think of it like a funnel. A bigger funnel lets in more water just like a bigger aperture lets in more light. The end result is a cleaner signal (to noise ratio, aka SNR).
Basically, if you have to rely on cropping AT ALL it's normally more effective to drop down a sensor size and get a smaller pixel pitch. It's why birders use APS-C and M43 cameras and not medium format or whatever.
But, if you can fill the frame the way you want it without cropping, and are willing to carry the gear and spend the money, bigger sensors produce better quality photos.