What "research" are you pointing to "proving" that three weeks of TPR is needed before starting storytelling??
There's no reason to require TPR before starting TPRS if you don't want to. TPR is useful to establish physical-movement words (though I personally believe there are limits to this as well, for the same reasons that the new "almost-kinda-comprehensible input" movement is not succeeding as well as TPRS: meaning is never established beyond a doubt because of the insistence that the native language is somehow "polluting" to the experience). But those words are easily established by simply telling students what they mean and then using them.
TPRS started as a desire to take TPR beyond the realm of commands. Yes, Asher claims you can teach the whole language through TPR, and I'm sure you can -- theoretically. I work with real teachers, though. The average teacher doesn't have the ability to do that beyond the very basics (stand up, sit down, touch your nose, whatever). Commands are important but they are not really the highest-frequency language in terms of what is on most curricula in a school course, and that also becomes an issue.
I start beginners all the time using nothing more than TPRS (and TPRS over Skype to boot, which limits some things). I make sure that the first items I teach are what I call the Super Seven: words that allow students to express and understand, although in a very basic and generic way, the concepts of possession, existence, location, volition, preference/affection, and motion. With those words, you can make some kickass stories, or talk about things that students really want to talk about. (Not all TPRS is stories about blue cats.) Then you go on to expand to the most specific vocabulary.
TPR is useful in a classroom situation because TPRS requires a lot of attention. It requires far more attention on the part of the student than traditional book exercises. They are constantly listening actively and answering and responding in TPRS. So using TPR to continue input while not requiring such close attention and oral responses is a good way to give a "brain break".
Associating gestures with new vocabulary items is very powerful, and is used by many (but not all) TPRS teachers, but that is not really TPR in the sense of Asher's stuff.