You are not exactly right, but you're also as far from being wrong as it's possible to be. Hitler, notably, was obsessed with gaining access to the oil of Chechnya, Dagestan and Azerbaijan, and in this (as in almost no other cases), he was reasonably correct. The Germans, by the summer of 1942, were heavily pressed for fuel, and they had exhausted almost all of their stockpile, and their sources (synthetics and Romanian oil) were substantial, but insufficient. The German summer offensive of 1942 is nevertheless regarded as something of a strategic failure, for basically abandoning the Moscow and Leningrad objectives, and because the infrastructure (or rather the lack thereof) in the region made the logistics (including, ironically, the delivery of automotive fuel) of mechanized operations difficult over the long term.

And so, Chechnya, Dagestan, and Azerbaijan were indeed major objectives of the 1942 offensive. The short answer to why the Germans didn't capture those oil sources (they did take Maikop, an important oil center, but the infrastructure had been destroyed beforehand by the Soviets) was simply that they failed to overcome Soviet resistance. The 1942 offensive, in addition to being imperfectly strategically conceived, was operationally a failure. If you look at the map you linked, you'll see a little hashed arrow pointing towards Kizlyar and the Caspian coast. That indicates the Germans' attempt and failure to do exactly what you (and they) said should be done. The Soviets won in 1942, and they kept winning, until they were in Berlin.

Additionally, during the 1942 summer offensive, Battle of Stalingrad began. The Battle of Stalingrad had many unfavorable antecedents for the Germans. At one point, you, the German High Command, and Hitler would reasonably have been in agreement about the priority of the Caucasian objectives. Originally, the entire Army Group South was tasked with, indeed, running straight to the Caspian shore. However, Hitler intervened and split that spearhead force literally in half, tasking one (Army Group A) with the Caucasus objectives, and one (Army Group B) with, you guessed it, taking Stalingrad. The half that remained for the Caucasus, as I stated above, was insufficient to the task set for it.

On the way to Stalingrad, the Germans encountered heavier-than-expected resistance, and were slowed, which bought time for the city's defenders. When the Germans arrived and began the battle for the city on August 23, 1942, taking the city became a slog that took all the rest of the summer, and it quickly escalated into both a titanic military clash and a contest of wills, prestige, and propaganda between Hitler and Stalin. As a result of that, some mechanized and mobile forces of the already-half-strength Army Group A were diverted to Stalingrad, at the same the Soviets were pouring new forces into the Caucasus from outside and bringing up local troops with better morale (especially Georgian mountain infantry).

So, that was kind of the plan. Fall Blau was supposed to go in four stages. The first two were basically positioning the German armies for a dash to Stalingrad. The German plan was to mask Stalingrad-- basically to make it irrelevant either by capturing it or surrounding it.

As you can see on this map the railroad link between the Caucasus and the rest of Russia ran through Stalingrad. That means that nearly all heavy equipment destined for the Caucasus would have gone through Stalingrad (we'll get to the sea lanes in a moment). That would have cut off shipments of heavy equipment.

But the sea lanes were still open, because Astrakhan (on the Caspian) was unoccupied at this point. Well, sort of. Those sea lanes were primarily used to ship oil. See, oil was extracted at places like Mozdok and then piped down to Baku, where it was refined and loaded onto tankers. Those tankers didn't necessarily stop at Astrakhan. They usually sailed up the Volga past Stalingrad. So, again, masking Stalingrad theoretically cut off the rest of Russia from its oil supplies.

So, while the Caucasus was still accessible, it was cut off enough.

And all that was before the actual planned invasion of the Caucasus. The final part of the plan-- Blau IV-- was intended to be the attack south. One part of the German Army would have gone straight for Astrakhan, the rest would have attacked the Caucasus directly. The ultimate target was the oil fields and Baku (which is so far away it's off the map). The full extent of Fall Blau is very unrealistic, because the logistical difficulties were just too great.

BUT! That's not why the initial stages of the campaign fell short of its goals. Hitler kept redirecting and micromanaging the armies involved. Postwar, a great many German generals liked to blame Hitler for their defeat. There's some truth to that here (though the German generals do protest too much). The most egregious mistake he made was to redirect the 4th Panzer Army away from the Stalingrad direction to the south to help the 1st Panzer Army capture the Caucasus. This left the 6th Army to push on to Stalingrad. It didn't have the strength to capture the city, but it was a near run thing. So, weeks later, Hitler weakened the thrust towards the Caucasus by moving the 4th Panzer Army back to the north to assist the 6th Army. But by then, defenses in Stalingrad had been beefed up (and fighting in a city with a panzer army was bad policy anyway).

This is a very broad overview, but I believe it answers the question.