This is my second reply. The first got removed. Let' s see if this meet the standard.

I'm assuming the video you talked if is either Epic History or Kings and Generals. They made great videos but for now, I want you to forget all the lauded generalships and think of these marshals are another set of humans you meet everyday.

I. Napoleon Plans and Friction

Clausitwitz who formed his theories based on the Napoleonic wars said to this effect "the generals' skills lied on the realm of chance and probability" which a great simplification is that they guessed. Before any campaign, Napoleon took great plans at making it. He had the maps ready, he invested heavily in intelligence gatherings and read the newspaper. (Chandler) You may heard "No Plan Survive First Contact With the Enemy". That is true for Napoleon, but he is a also master of improvision. Auerstadt were a planning failure. (Cook)

Clausiwitz had contribute another concept in strategy called "Friction" to explain the plan on paper and reality. People can be late. Rain may messed up. The maps are wrong,...etc.

In this controversial incident, the short answers normally found are 1. The roads are bad, and became worse due to rain. Found inside a book sympathetic to Bernadotte. (Palmer) 2. Bernadotte is jealous that Davout, a younger marshal, is supposed to lead him. Found in book hostile tp Bernadotte (Chandler)

Longer answer is in this link: https://www.napoleon-series.org/military-info/battles-campaigns/bernadotte-1806-is-there-a-case-for-the-defence/

In other words, the events in this battles is nothing more than "Friction" at play. Napoleon' s plan was wrong. Despute being a great man-manager, Napoleon is a type of boss who blamed their underlings for when things when wrong. (Chandler) This event would have been just a normal event in those wars if not for Bernadotte and Davout achievement and feuds.

II. The Gascon and the Iron Marshal

Now, who is Bernadotte? "In an era with no shortage of controversial men, Bernadotte is one of the most controversial" (Gill).

David Chandler, one of the most influential historian of the Napoleonic era, did not view Bernadotte positively. In the War of the Sixth Coalition, Chandler called him a "coward". Many sources, especially Prussian, described bitterly of Bernadotte bloated ego and languid pace. (Chandler)

Chandler' s assessment never seem right with me. Bernadotte often fought on the frontlines recieving injuries (Palmer). He is also personal friends to many marshals. Marshal Ney used to be under his command so did Grouchy and many other talented generals such as Vandame or D' Erlon. To say that Bernadotte is jealous is a facepalm. They hate each other, but not enough to screw up the wars.

As for Davout. "One of the purest glory of France" (Napoleon).

Davout' s great military success came from the intelligence network that he built. (Gill) His many spies are what made other marshals hated him, not only because he is a no-nonsense general. Once, he even read Bernadotte wife letters in front of his face. (Barton) Massena, Lannes, Vandamme, Berthier, also found themselves despising this general who is also one of Napoleon finest. (Gill)

Bernadotte, in contrast, is highly popular and became a political rival to Napoleon. When Napoleon had his coup, Bernadotte do not know if he going to be alive, dead or exiled. He caught Davout spy in his house. (Palmer) Imagine how he felt.

So when Davout were facing an army much larger than his, no help arrived. He may believe that the gascon used the opportunity to get rid of him. The infightings of Napoleon marshals are legendary. I.e. Lannes asking Soult for a duel on the preparation of Auesterlitz. Ney and Oudinot messed up the Prussian campaign. (Barton) Davout and Bernadotte feud again is one of many. Napoleon' s marshals has a distrust toward each other intention. Bernadotte think that Berthier framed him. Davout believed that Bernadotte let him to be in danger. (Palmer)

III. The Event and its fall Out

Napoleon spread out his corps telling them to go to different areas and await further instruction. When scout updated that the main enemy might be in Jena, he concentrate his available forces on it. (Cook)

Bernadotte and Davout are travelling to the destination they were instructed to and too far away to affect the battle. When the plan changed, Bernadotte were moving his army across a terrible road (Palmer) and Davout is fighting the main Prussian army, Napoleon believed to be in Jena.

Due to Davout mistrust of Bernadotte, he blamed him to intenionally not moving to the sound of the guns. (Barton) . The sounds of the guns arguments are far fetched. If Bernadotte can hear it from Auerstadt, he can also hear it from Jena. If Napoleon is to fire Bernadotte for this incident, he would brought up how his miscalculated the campaign (Cook).

Bernadotte' s later success brought down the Napoleonic empire, making him a target for character assassination from soldiers more faithful. Fans of Napoleon questioned his integrity and personal capabilities. This event is proof it. But examining the events based on what the generals act on their knowledges, it is another "friction" which happen regularly in the Napoleonic wars. It would have been forgotten had Davout did not made his name and Bernadotte did not became a king.

If Napoleon fired Bernadotte, he would able to rid homself of a former rival but have to deal his older brother, who sworn to protect him and the many friends he had in the army ranks, and removed one of his most capable marshal at the time. (Barton)

Sources:

P. d. Barton. "The Amazing Career of Bernadotte".

Alan Palmer. "Bernadotte: Napoleon' s Marshal. Sweden King"

David Chandler. "The Campaigns of Napoleon"

John Gill. "1809: Thunder on the Danube"

John Cook. "Bernadotte 1806 Is there A Case for Defense?"