What are the origins of such a push towards Kalinin, away from Moscow?
The answer can be found in a book by Jack Radey and Charles Sharp, The Defense of Moscow 1941, The Northern Flank.
From page 4, bold is mine:
On October 7th, the same day that the Soviet forces in and around Vyazma were encircled, the German Army's commander in chief, von Brauchistch, met with the commander of Army Group Center (von Bock) and discussed changing the original plans. Specifically, while the Army Group advanced on Moscow with everything that could be spared from holding the encirclement ring, he proposed also advancing 9th Army and Hoth's Panzer Group 3 'in a northwestern direction' to clear the northern flank of Army Group Center.
There is a recurring topic in the German conduct of war at this stage of the invasion of Russia: the total focus in destroying enemy forces. Noted the word "clear" in the previous paragraph?
In another paragraph, some two pages later, the author's note that Panzer Group 3's new objective (clearing the northern flank) was shared with units from Army Group North (!).
Was this ugly north flank the result of just the eagerness to encircle and destroy as much Russian armies as possible? Or was there also a need to have a decent left flank for the advance towards Moscow? In other words, could Army Group Center afford to send Panzer Group 3 directly east towards Moscow instead of this push in a northwestern direction?
I don't know. Playing the Panzer Campaigns Moscow '41 monster scenario (the one that covers the whole operation) could provide some educated guesses. But I can't afford to embark in such a huge scenario due to real life obligations.
Another possibility is playing the Moscow 41 scenario in Gary Grigsby's War in the East. The screenshot below is how the scenario looks like. I included the historical axis of operations for Panzer Group 3 (green arrows).