I suppose we need to have a semantic discussion about what exactly "full of" means. Note that the original person who started the thread used the phrasing "many" (which only moves the semantic discussion to how many artifacts with dubious origins you need to have to speak of "many").
Personally I think that's not a very interesting discussion. Your general point that there are also a great many artifacts are not stolen (the majority): yes, you are correct. And I agree it's worth pointing that out as some people do overstate the case at times.
Not the marbles, but the "British Museum is full of stolen" stuff is a meme age thing and not true (go there and check the origins of what you see).