I would agree with all you have said as well as add:
There are many types of Body Odor causes so it is a process of elimination. The best thing to do IS to cut out all the crap food, and start eating fresh fish, grains, vegs etc.
Have a good clear out of the colon. Dr Natura Colon Cleanse and the Toxic removal system are supposed to be excellent..google it
Treat your mouth, start oil pulling and use ORAMD instead of toothpaste, again google it
This has helped my mouth stay fresh, my gums are looking much healthier, tongue is pink and toothpaste now tastes of suger and chemicals! never go back to toothpaste.
I dont sell ORAMD (see no link) I just use it.
This will not be a sprint, it wont happen in a day, esp if you have had the problem for years or have particularly bad odour but YOU WILL GET THERE, you have to make sacrifices though, it would be great if you could eat onions and not have stinky breath for days afterwards but some of us are just not built that way.