Modern cars minimize soot mostly by carefully controlling the combustion to produce a minimum of it. Any unburned fuel is an inefficiency, reducing gas mileage. Computer-controlled fuel injection helped advance this a lot (over carburetion).
Most cars using a diesel engine are indeed fitted with a particulate filter nowadays. It is more or less necessary to meet environmental standards in most of the world. So at least some cars do.
Cars with modern fuel-injected engines produce very little particulate exhaust, and what little they do produce is burned in the catalytic converter.