Were the Germans in a position to capitalize on their military objectives at Stalingrad without capturing the whole city?

Fall Blau objective was to control oil fields of Baku, Maykop and Grozny. Stalingrad was secondary to that. But, as usual, german plans changed during the execution of the plan, making Stalingrad the main objective after a while.
Now, as you propose, if we imagine that they keep their original plan, when Stalingrad was secondary. Control Volga river would prevent Soviet Union to use this river to transport oil from Caucasus.

But

Soviet Union already had other sources of oil. Second Baku was already discovered and under development. Also, the allies were able to send oil through Iran and the Pacific. Obviously, infraestructure was required to make use of this alternatives.

Leaving the Soviets in possession of part of the west bank of the Volga within the city would mean that the Germans would not have a secure front. The Soviets would have been capable of supplying and reinforcing their troops via small boats at night. That would allow them to keep the front within the city active, soaking up German manpower and supplies.

The German problem was manpower. If the Germans held the whole of the west bank within the city, the Soviets would have had a harder job attacking them across the river, and the Germans would not need as many men to hold that line as they would to hold a line within the built-up area.

If the Germans had given up and pulled out of the city, they could have held a line on the plains with fewer men than they'd have needed for a line in the built-up area, but probably more than they'd have needed for holding the river line.

This is basic military calculus of forces, obvious to trained officers, which is probably why few books mention the issue. Once the Germans had started trying to take the city west of the river, they needed to finish the job, or give it up and pull back out of the city. The problem with pulling out was political: it would have looked like a Soviet victory.

Frame challenge. The German positions, even at deepest penetration, were undefendable.

...the Germans held portions of the west bank both north and south of the city on which they could place artillery and air units to disrupt Volga River traffic.

Was there any time prior to the encirclement that the Germans' military objectives would have been substantially fulfilled by consolidating their holdings and "ring fencing" the remaining portion of the city while they wen about their business (like setting up a defense)?

The easiest part first, there were no airfields that far forward that were useful (or, indeed, useable) to affect Volga traffic. The airfields that could be used were the same that were later used to (attmpt to) support the Stalingrad pocket, 50 or 100 kilometers back. You don't want to rearm bombers on the runway while in range of the enemy's artillery. So airfields were a non-issue either way. What about ground troups positioning?

North of the city, a small corridor of about 10 kilometers width was held since late August 1942 by German XIV Panzerkorps. This did interrupt the railway lines from Stalingrad to Moscow; that much was achieved. However, this corridor was through open steppes. The Don (which further west provided a defensible landmark for the Romanian 3rd Army curves to the south of Stalingrad.

This northern corridor was under continuous attacks from the north by Soviet 66th Army, 4th Tank Army, 1st Guards Army, and 24th Army. These attacks were of a magnitude that XIV Panzerkorps was unable to assist with attacks on Stalingrad. Actually, LI Armeekorps' advance on Stalingrad from the West was temporarily halted, its support redirected north, to reinforce XIV Panzer Korps.

Only two to three weeks later (around September 11th) German forces were able to close the pocket around Stalingrad in the south. However, this corridor (held by XXXXVIII Panzerkorps) was also under significant attacks by Soviet 64th Army from the south.

All the while, the units of Soviet 62nd Army fighting in Stalingrad proper were getting reinforcements from the eastern bank of the Volga.

Any German units on the west bank of the Volga -- be that in the northern corridor (XIV Panzerkorps), the southern corridor (XXXXVIII Panzerkorps), or in the city proper (LI Armeekorps) -- had Soviet forces on two sides (not counting artillery fire across the river). Quite close as well, no more than 5 kilometers at the most. Not the position you would want to dig into for defense, let alone bring your artillery into. Especially since the terrain outside of the city is nothing but flat steppes

Note that 62nd army was continuously reinforced across the river, which should give an indication that river traffic was not really "controlled" at that point.

German troops were being attacked on all sides. There was no position that could be made into a viable defensive line that could be held through the winter. As Operation Uranus proved in November, which did not target the German spearhead at all, but its exposed flanks hundred miles and more to the west and south.

To summarize.

The advance on Stalingrad was a spearhead, its grip on the west bank of the Volga tenuous, its flanks under massive pressure. Winter was fast approaching, which the Germans had learned meant the upper hand for the Russians for a couple of months. Those encircling corridors did little toward the military goals (other than interrupting the railway to Moscow), could not have been extended to do more (that is what the Germans were trying), and could not have been held as-is during the winter.

A great week by week visualization of the Battle of Stalingrad is this video by Indy Nidell. Watching it should make it clear how tenuous and short-lived the hold to the Volga was.