They tried...or at least some of them dreamed of trying.

In his memoirs, Rommel talks about his grand idea of beating the British in Egypt and breaking through into the Middle East, both to access the oil there and to cut off the quickest route to India from Britain.

There were obviously a lot of problems with this dream of Rommel's, though. For one, as we know, he was unable to beat the British in Libya/Egypt, largely due to the British presence in the Mediterranean Sea making his supply situation a nightmare -- a nightmare that got worse the further east he advanced. So that issue would have to be resolved before he could realistically advance very far into the Middle East.

Which brings us to our second major issue, which is that the German higher ups were not in agreement with Rommel's dream of conquering the Middle East. The never gave Rommel as much as he wanted, and largely viewed the North Africa campaign as a sideshow to the main event in the USSR.

The final major issue is basically the first issue, but in reverse. If Rommel was somehow able to conquer the Middle East, the Germans would then have to figure out a way to get all that oil back to Germany. The conquest of the Middle East would make the Royal Navy's situation in the Mediterranean much more difficult. But even if we're generous and say that such a conquest would leave the Axis in almost complete control of the Mediterranean, there's still the matter of having the shipping to move all that oil.

As a contrast, that oil in the Caucasus could be taken by land to Germany, in many cases using existing pipelines that already existed to transport oil in the general direction of Germany.