Well, I guess that could be done, but I do not think it would be the same as the quite a bit of smoke penetration I get while cooking and the smoky bark I get from the entire smoking process. Only sometimes I like a little more smoke, so add it at the end after pulled or sliced. Most of my meat gets packaged and frozen for meal sized thawing and reheated later so there is time for the added smoke to redistribute. My Smoke-It brand of electric smoker has very little air flow 3/8 hole in the bottom, 1/2 at the top and is sealed very well so I do not think a smoke tray or tube would stay lit. I have opened and added little wood chips during the smoke also at times which helps also.
I feel my method gets some smoke flavor all the way to the middle of the center meat of a large chunk like pork butt after pulling which may not happen with all smokers.
And along your line of thinking, I do frequently mix up a pan of chex mix covered with a coating of butter and seasonings to put in the smoker till it is back crispy again adding smoke, which is similar to your statement. Yep, if enough of us throw out what works for us, hopefully he will find the best way to find what works for him. Thanks to all out here and Merry Christmas.