If we accept that Hitler was focussed on Stalingrad for political reasons it doesn't explain why he wanted to do this by pouring troops and resources into the meatgrinder that Stalingrad became.
Even without hindsight it seems fairly obvious that it would have been possible to "siege" Stalingrad while pushing forward elsewhere. Only after having pushed the Russians back elsewhere.
If, in Hitler's mind, capturing Stalingrad was the only option was there not still better ways to achieve this? Why did Hitler feel a direct push into the city was preferrable to trying to encircle it for example? I understand his apparent obsession with the city but not the way that goal was worked towards.
Stalingrad became important for several reasons.
The first and foremost reason was that it was right on the Volga river, a major transportation hub for the Soviets. If the Germans managed to cut that line of communication, the Soviets would have a very hard time to supply their forces in the Caucasus. And without those forces, the oil fields there would fall into German hands - a major Soviet force and a vital source of raw materials would fall to the Germans. Such a fall could also cause geopolitical consequences - the German ability to influence Turkey would increase, German forces could perhaps move to interdict the Anglo-American lend-lease supply lines through Persia and inspire the occupied Persians to revolt against the British and Soviet occupation, limiting the amount of oil the Soviets and Allies could extract from Persia and requiring both to reinforce their troops there at a vital time in the war.
As you can see from this map the railroad network east of Stalingrad was very limited and far away. The Volga was simply a very important part of the Soviet infrastructure at the time.
Stalingrad itself also housed several large factories that were supplying the Soviet forces in that part of the Soviet Union with trucks, rocket launchers, AT guns and T-34 tanks from the Red Barricade, Stalingrad Tractor, Dzershinzky Tractor and Red October factories - these continued to produce arms for the Soviet defenders even as the Germans were assaulting one end of the factory.
Why continue to push Stalingrad? It was even more a meatgrinder for the Soviets, it was a vital communication hub and offered the chance to cut the Volga river as a supply line for the Soviets.
Stalingrad lies at the bank of the Volga river and the Soviets had a major supply depot on the other side of the river, so it was not possible to lay siege to the city - the Soviets could reinforce and supply the defenders by barges and ships crossing the river at nighttime - this is also one of the reasons the Germans continued to push. If they had secured enough parts of the west bank of the Volga river, they could have mounted searchlights and guns to destroy the Soviet nightly supply convoys across the river and thus force the substantial Soviet defenders to surrender. Crossing the Volga river was not considered doable at the time - especially as the Soviets still held substantial forces in the city and could use them for a counter-attack.
Leaving Stalingrad as a major Soviet bridgehead would also threaten the flank of Heeresgruppe A that was advancing into the Caucasus at the time.
So, Stalingrad was an important strategic point, an important industrial center, offered the Germans the ability to cut the important Volga supply route for the Soviets, held substantial Soviet forces and its fall could probably have affected the Soviet forces in the Caucasus very negatively, which could have substantial strategic knock-on effects.
It was an important target, and a single German army (with some support from another) out of the six armies and reserves of the Heeresgruppe B probably looked worth it to the Germans at the time.
