By the time of the Ottomans, Iraq had devolved to a nomadic backwater.
It's heyday in the Abbasid times had long past. Starting it's decline in the Anarchy at Samarra in 860s. Then the failure of Muqtadir and his successors to maintain the irrigation system. Added with the climatic catastrophe from 920-1060 and the Seljuk invasions, Iraq was a husk, agricultural cultivation at one of its lowest points in 5 millennia, only Baghdad clinging on. The Mongols snuffed out Baghdad, though they did somewhat rebuild it after it's destruction. Until finally Timur dealt Iraq it's last death knell. After which it became almost entirely nomadic...
It was only under the British in the 20th century, that large scale Irrigation began to take place again. Restarting agriculture in southern Iraq after a 1000 year hiatus. But the millennia neglect left its toll, with large swathes of southern Iraq (Sawad - formerly the birthplace of civilization and one of the most fertile regions on earth) being uncultivatable salt crusted desert. With majority of modern Iraqi agriculture is focused in the northern rainfed regions of the country.
For the Ottomans the main point of it was as a bulwark against the Safavids. As well as the importance of Basra as a node for Indian Ocean expansion. But overall it was a backwater.
The lack of Iraqi prosperity made it an unenticing target for the Safavids, though there was a strong religious aspect. But the political instabilities of the Safavids (unlike Parthians and Sassanians) made them unable to carry out a campaign into Iraq even if they wanted to. Except in exceptional situations when they had strong rulers like Shah Abbas or Nader Shah.
Whereas in the time of the Romans southern Iraq was probably the richest region of western Eurasia rivalling Italy, since it seems to have produced 4x the wealth of Egypt (at least in time of Umayyads - prob more in Sassanian times with smaller marshes)
Beyond that, it was the Persian's only major bankroller. The Iranian plateau not being particularly conducive to agriculture having large nomadic populations and Transoxiana being only tenuously controlled and not nearly as prosperous as Mesopotamia.
It was the site of their capital, the source of their wealth and had been one of the densest and most advanced regions of the world since it invented civilization. So they fought tooth and nail to maintain control over Iraq. Even if Romans temporarily took Iraq, they couldn't hope to penetrate to enter Iranian plateau due to logistical difficulties and superior Persian cavalry. So the Persians, waiting just beyond the Zagros, could easily assault the precarious and overextended Roman position in Iraq and retake their heartland.