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With the right muffler, these FE's are as quiet as any V8.
I think I know what the OP meant. I'm still trying to find a "quiet" muffler. When I got my truck about a year ago I replaced the glass pack on it with a "turbo" style muffler since that's all the NAPA guy showed for it. I finally got around to replacing my leaky exhaust manifold gaskets (put on heat shields) last week and ordered a "Walker Quiet-Flow SS 21151" muffler to replace at the same time. Well, the Quiet-Flow is not "quiet". It's just as loud or maybe louder than the cheap turbo type. Rockauto showed the Walker 21151 to fit my truck though Walker's website shows no replacement (69 F250 w/360). So even though the muffler has a 90 day guarantee (no matter what), Walker won't honor it because they don't officially show it to fit.
So, I'll just live with it a while and find a quiet one someday.
These trucks have so much room you can fit just about anything you want. Simple logic, the most quiet system will be stock cast iron manifolds and a single stock muffler - simplest scenario and very little maintenance required. Your truck has headers, they are a compromise in that you get some better performance in exchange for added performance and some extra work replacing gaskets that will blow out. Headers are a "you get what you pay for", higher quality sets have thicker flanges and better collectors. As for the mufflers, anything will fit from stock styles to turbos, glasspacks, etc. as long as the pipe sizes are right.
One of my trucks has a dual 3" setup with is loud at idle and even louder under throttle but I like the sound. I also have a completely stock system on my daily driver, it will take me anywhere I want to go nice and quiet.
Your short video clip sounds like you need new manifold and collector gaskets or may even have a damaged/rusted thru tube. You can feel around the tubes (but not touch!) while it's idling to see if you feel the leaks. It sounds like it will be pretty obvious.
It's an old school V8, not a wheezer 4 banger like your neighbors have prolly. With some exhaust leaks. Time for header gaskets, duals if the pipes are bad, and mufflers of your choice. I'm a fan of Cherry Bombs but Garrett (351clevelandC4) has a sound bite of his exhaust system on one of his threads. Sounds mucho bueno.