If you want to understand just how much the exhaust system does, try driving a vehicle with bits of it missing...
The abovementioned mechanic did LPG conversions and sometimes there would be a need to alter the path of the exhaust system to get around the LPG tank. I was the one usually chosen to drive these vehicles over to the exhaust place - and let me tell you, you get a REALLY good first hand lesson in the accoustic output of even the most moderate engine - especially when under load. As a result, my driving in these situations was extremely conservative - but there was still a bridge over a railway that had a steep approach and the noise was intense.
Sometimes the only bits of the exhaust still attached were the manifold, engine pipe and catalytic converter. BIG noise! (and, yes, I was constantly looking in all directions for any police interest - which I never encountered, thankfully)
Had my exhaust replaced recently, had to limp around town with nothing on while they were getting parts in. Even the weedy engine in my Toyota sounds like a hotrod* when it's loud and open like that.
*A hotrodded lawnmower, to be more honest / accurate.

Also quite a difference; the previous system I had, was welded from pipes and parts. In particular it was missing a mid-line resonator. Never quite sounded right. (This time around, it's all aftermarket or OEM parts, close enough to original. Sounds about right.)
Also a little surprising how much, and how little, difference can be made from little leaks here or there. Rusted out pipes tend to get loud, and add a rougher sound with more hiss (modulated hiss -- turbulence on the pulses). Sound from underneath or behind the vehicle tends to carry pretty well. OTOH, leaks in the engine bay don't seem to do much, presumably as the sound is forward-facing and reasonably well absorbed and scattered by everything in there. (They weren't able to seat the downpipe quite right so I had this going on, before getting it re-checked.)
Tim