Contrary to popular belief, before they declared their support for Israel, the United States was viewed favorably by many Middle Eastern peoples. They were seen as nation that had overthrown their colonizer (something that deeply resonated with the Middle-Eastern nations).

This relationship was born after WWI, when Wilson's 14 points of self-determination appealed to the Arab Nationalists seeking independence from the colonisers. For example, the Wafd Party (an Egyptian Nationalist party that originated in the early 20th century) tried to convince the British to let his delegation make their case in either London or the Paris Peace Conference. Unfortunately for them, they failed and Egypt remained under colonial rule (via protectorate) until the Free Officer led coup in 1952.

This relationship then popped up again with the propaganda leaflets dropped by the American forces when Operation Torch began in North Africa during WWII. The translation is on the page I linked to and you can see that it focuses heavily on certain rights - the right to worship, live in peace, etc. As can be seen, these echoed the 14-Points of Wilson.

So why did this start to change? In many ways, this was due American support of Israel in the 1948 Israeli War of Independence. Additionally, many Arabs felt betrayed at the United States not pushing hard enough for their independence. With the successive Arab losses to Egypt, particularly in 1967, Arab thought can be seen to shift. This is the time when the key thinkers modern Islamic/Islamist groups began to appear. Hassan al Banna's Muslim Brotherhood collaborated with the aforementioned Free Officer coup in Egypt (before being brutally cracked down upon) and from a splinter of this group arose Sayyid Qutb, who can be seen as the ideological father of modern Islamist Terrorism. Much of the anti-American sentiment in the Middle East can be seen as originating around this time.

However to get back to your question of whether the Middle East has always been so unstable we should look at the history of colonialism in the region. The Middle East in the 18th century was largely unified by the Ottoman Empire. However, throughout the 19th century and until its fall in 1922, this Empire was in decline. Specifically, the key Northern African territories were lost throughout 19th century - with Algeria falling to the French in 1830, Tunisia to the French in 1881, Egypt to the British in 1882 and Libya to the Italians in 1911. From that point on, a system of protectorates and direct colonial provinces were established. Then, after the fall of the Ottomans, the well-known Sykes-Picot treaty divided up the lands from Palestine to Iraq, handing them over as mandates to the French and British. The division of territory, although somewhat following traditional Ottoman provincial boundaries, give little regard to various ethnic or religious groups and at times actively played them off of each other. The remnants of this can especially be seen in the rule of Bashar al Assad in Syria today. The Alawi minority of the region was given greater powers (particularly within the Army) as the French believed that they would be less likely to risk their own preferred status by overthrowing the system. This privledged minority idea has continued within Syria to this day as Assad (an Alawi himself) has tied the fortunes of the entire ethnic group to his regime.

This is not to say that the land was entirely peaceful before the Europeans arrived; the 1860 massacre of Maronite Christians in Lebanon in particular comes to mind. However, many problems were exacerbated by this colonial presence. The economies were set up to benefit the European colonizers at the expense of the native inhabitants, Europeans were given greater rights (the disastrous effects of this can be seen in the Algerian war of Independence, when the Algerian National Liberation front killed thousands of European settlers and collaborating locals), and there were brutal repressions of civil rights in order to further economic gains.
In this way, the region could be viewed as being "in a poor state prior to USA engagement" as you have said, as it was other Western powers who worked for their own benefit.

So is the United States fully/majorly responsible for instablility? I think there is blame to be shared. The United States was not innocent in the region, however to call them the cause of the modern problems would be a bit far. The colonial legacy and religious and Nationalist tensions, compounded with the inequality and rapid modernization caused by the discovery of oil, have had a grand impact on the region.

Now one could talk much more about the impact of economic or societal factors (in particular Arab Nationalism and Arab Socialism have had interesting impacts on the region, as well as religious ideas tied to avoiding fitnah (discord)) leading to a succession of authoritarian states instead of democracies however that is a different discussion. I hope I at least started to answer your question! This was rather general but if you have specifics I can do my best to answer, and I am sure there will be others with detailed responses.

Sources

A History of Islamic Societies by Ira Lapidus

The Looming Tower by Lawrence Wright.

Various general ideas taken from assorted articles and lecture notes.

EDIT: Some clarification on sources. In no particular order.

State, Power and Politics in the Making of the Modern Middle East by Roger Owens. In particular chapters 4 and 9.

"The Robustness of Authoritarianism in the Middle East" by Eva Beilin (especially for any questions about the factors I mentioned in my final paragraph)

Napoleon in Egypt: Al-Jabarti's Chronicle of the French Occupation, 1798

Le choc colonial et l'islam by Pierre-Jean Luizard

Portrait du Colonisé by Albert Memmi