Having inadequate, plugged, or partially plugged vents can be demonstrated by example. Stick a straw/drain pipe into a glass of water and you can pour water down the straw/drain pipe. Put your finger over the end of the straw and pick it up and the contents of the straw/drain pipe will defy gravity. Do the same experiment in slushy or crushed ice and the slush/crushed ice will pack off and plug up just like sewage and paper from a toilet will when your plumbing isn't vented properly. Just food for thought to help you understand some of physics/dynamics behind plumbing codes and also help explain why all the diy plumbing books you have been looking at have the same basic way to install traps. I can't see your system but a plumber can look at it and maybe design a system to avoid too much cutting into the walls and the cutting you do have to do may not affect the structure that much.