I am not sure because, eh, I'm not using lower end devices, but it seems that the most reasonable way would be the browser to check for the capabilities declare

I am not sure because, eh, I'm not using lower end devices, but it seems that the most reasonable way would be the browser to check for the capabilities declared by the graphics drivers and then decide if they offer the option of background blurring. I don't think doing it server side would be cost efficient.