The only Panzers that reached Normandy on D-Day itself were the 21st Panzer Division. . . which was pretty much annihilated in short order, losing 70 of their 124 tanks. They did prevent the British from taking Caen, but utterly failed to "drive the invasion into the sea". Rommel's other two Panzer divisions, the 2nd and the 116th, were held north of the Seine specifically to respond to Calais, and thus were too far away and out of the equation entirely. The 2nd SS, 9th Panzer, and 11th Panzer were also out of the equation, having been reassigned to Army Group G in southern France. And the four remaining Panzer divisions of Panzergruppe West -- the 1st SS, 12th SS, 17th SS Panzergrenadier and Panzer Lehr -- were held as reserves that could only be released by order of Wehrmacht Supreme Headquarters
Thus, Rommel's fundamental defensive plan of having all ten Panzer divisions in the theater stationed for an immediate and decisive counter-attack was made impossible. While Supreme Headquarters did release the Panzergruppe West reserves, "quickly followed" is totally inaccurate -- the orders came late and 12th SS was the only reserve unit to arrive on D-Day, and even then only the lead elements arrived by nightfall. Despite being one of the first to move, Panzer Lehr was not fully in position until the 9th. 2nd SS was ordered to Normandy immediately, but it too took days to arrive. 17th SS arrived on the 11th, 2nd Panzer on the 12th, 1st SS days after that.
The Panzer divisions were able to inflict significant delays and losses on the Allies, but the war was fundamentally lost to Germany by the afternoon of D-Day.