Until it died down. Is that the true story?
Not at all. "Just threw men at it" is perhaps a fair description of the German effort to take Stalingrad. The Soviet plan was both more sophisticated and more effective. They fed the minimum reinforcements needed to keep the fight for the control of Stalingrad going, to keep a foothold in the city. Rather than throwing men into the battle, they trickled just barely enough men into the battle.
Soviet success at keeping a foothold in Stalingrad meant that the Germans were denied victory. The rubble-choked urban environment gave Soviet forces plenty of places in which to hide, and made for fine defensive positions. German mobility was limited. The Soviet defenders reduced the effect of German firepower, in the form of artillery and air support, by keeping their positions as close as possible to the German lines. The Germans were unable to fight a war of maneuver, and were forced into a grinding battle of attrition in their effort to take the city. Their failure to take the city forced them to commit more men to the battle.
The Soviets had plenty of men who could have been fed into the battle in Stalingrad. Where did they go instead? Three Soviet armies were concentrated north of Stalingrad: the 5th Tank Army, the 21st Army, and the 65th Army. This was almost 300,000 troops and over 600 tanks facing the 3rd Romanian Army, with a strength of about 150,000 and weak in tanks and anti-tank weapons. To the south were the Soviet 51st Army and 57th Army, about 100,000 troops and over 400 tanks facing the 4th Romanian Army, with a strength of about 75,000 and, again, weak in tanks and anti-tank weapons.
The flanks were weakly held - German forces had been pulled into the fighting for Stalingrad, and the flanks needed to be defended by what forces were left. Worse, the Soviets still had bridgeheads across the Don on the northern flank, depriving the 3rd Romanian Army of a more easily defensible line along the river. The fighting for Stalingrad had been given priority over eliminating the Soviet bridgeheads across the Don.
While the outnumbered Soviet forces in Stalingrad fought hard to slow the Germans, and to keep their precarious toehold in the city, the forces noted above - over a third of a million men and 1000 tanks which had been kept out of Stalingrad - punched through the weak Romanian armies on the German flanks and trapped the bulk of the German 6th Army in Stalingrad. While the Germans had committed themselves to a grinding battle of attrition in Stalingrad, the Soviets had the vision to aim for a bigger prize. Rather than simply defend Stalingrad, they opted to leave Stalingrad weak, and lured more and more German forces in. They then sprung the trap, and caught the German 6th Army. This operation (Operation Uranus) demonstrated the growing strength and skill of the Red Army, and the vision of Soviet high command. The Germans had been thoroughly out-Blitzkrieged.
This was not the end of Soviet ambition. While the German 6th Army slowly cooked in its cauldron by the Volga, the Soviets aimed for even bigger prey: the German Army Group A, consisting of 3 German armies. Operation Saturn aimed to trap Army Group A in the Caucasus. The Soviet plans collided with Operation Winter Storm, the German attempt by 4th Panzer Army to relieve Stalingrad and save 6th Army. Operation Winter Storm, which caught the Soviets by surprise, forcing the Soviets to deal with the immediate problem of the German attack. A modified Soviet plan, Operation Little Saturn proceeded a few days later, first stopping the relief force (only about 50km from the Stalingrad pocket), and then attempting to trap it. Operation Winter Storm lasted for less than 2 weeks before the Germans were retreating back to their start lines to avoid encirclement.
Next, the original goal of the trapping Army Group A proceeded. Again, the German flanks were weakly held by allied forces (again, weak in armour and anti-tank weapons). The Soviet offensive annihilated the Hungarian 2nd Army (which ceased to exist) and shattered the Italian 8th Army (which lost about 2/3 of its strength), and Army Group A was only saved by prompt abandonment of the Caucasus.
Soviet victory at Stalingrad was not due to throwing men into the battle in the city. The fate of Stalingrad was not decided in Stalingrad itself, but on the flanks over 100km away. The Germans were not simply stopped at Stalingrad (they had early been stopped at Moscow); an entire German army was trapped and destroyed.
The Soviet deployment for Operation Uranus, the encirclement of the 6th Army at Stalingrad:
The success of Operation Uranus and Operation Little Saturn:
The light yellow ("to 12 December 1942") shows the encirclement of the 6th Army. Next, "to 18 February 1943" shows the gains of Operation Little Saturn.
No, Enemy At The Gates is not a documentary, blocking detachments did not machine-gun retreating troops, nor were untrained conscripts sent into battle with one man getting a rifle and the other getting the ammunition; all of the above is tired old German-Soviet War myth, as explained by u/Georgy_K_Zhukov.