To begin with, the Abbassids did not claim descent from Muhammad. The Sunni definition of Ahl al-Bayt is broader than the Shi'a definition. Sunnis consider all of Banu Hashim and their descendants to fall under Ahl Bayt. That would be the descendants of Ali, Abbas, Ja'far, Aqeel, and Harith ibn Abdul Muttallib. The definition is even broader in the Shafi'i school. The founder, Muhammad ibn Idris ash-Shafi'i, considered Banu Muttallib and their descendants to also fall under the definition of Ahl-Bayt.

Shi'as, of course, consider Ahl Bayt to only refer to the descendants of Fatima. Even Ali's children through other women are not considered Ahl Bayt.

So the Abbassids, being descended from Abbas, are considered Ahl Bayt by Sunnis but not by Shi'as. An important point to note.

To answer your question.....kinda. As with most historical events, it's too simplistic to point to one cause and say it was the only reason. But yes, Umayyad hostility to Ahl Bayt played a role. In particular, the incident at Karbala was a major stain on the Umayyad reputation. It fomented rebellion and caused a near continuous string of uprisings between the time it occurred and the time the dynasty ended. Proto-Shi'as attempted many times to overthrow the dynasty and install an Alid ruler (descendant of Ali). The Abbassid's capitalized on this and allied themselves with the Shi'a movement.

However, realize that the proto-Shi'as were a relatively small movement. Alid sympathizers were larger and included many of those who would be considered Sunni. One of the most famous Sunni jurists, Nu'man ibn Thabit (Abu Hanifa), was pro-Alid and contributed financially to one of the rebellions led by Zaid ibn Ali (a grandson of Husain, the one killed at Karbala).

Even larger than just the proto-Shi'as and Alid sympathizers, however, were non-Arabs. Under the Umayyad dynasty, the majority of Muslims were non-Arab. However, there was heavy discrimination against them. To the extent that at times, jizya, a tax taken on non-Muslims was taken on non-Arab Muslims. This was more than just an issue of money, it was the highest insult you could give a Muslim. Basically saying that their Islam wasn't quite as valid as an Arab's Islam. What was particularly infuriating is that by this time, the religious and scholarly class was majority non-Arab. I mentioned Abu Hanifa earlier, he was non-Arab. If you go through a list of the most influential jurists, traditionalists, and other scholars, it would be dominated by non-Arabs (mostly Persian). Even if you weren't a scholar, imagine how angry it would make you to see that the most religious people in society are non-Arabs and yet non-Arabs are treated like second-class Muslims.

Abbassids made use of this. They organized their military units not by tribal affiliations or ethnicity but by location. It didn't matter if you were Arab or non-Arab, you were grouped by where you came from geographically. The modern equivalent would be replacing "White" regiments, "Black" regiments, and "Asian" regiments with a "New York" regiment, a "California" regiment, and a "Kansas" regiment. Not only does that show your supporters that you're not preferring a certain ethnic group or tribe over others, it creates a new solidarity with the people around you.

It's said by some historians that the Abbassid revolution represents a shift from an Arab empire to a cosmopolitan empire. That is true in a sense, but remember that the Abbassids were still Arabs, their language was still Arabic, and their culture (while Persian influenced) was still Arab. It opened up the doors for non-Arabs to join the political leadership but it wasn't a Persian defeat of Arabs as some people think.

All of this to say, the racial issues of the Umayyads played a bigger role than proto-Shi'a sentiment. The Abbassids benefited from Shi'as aligning with them but it's unlikely they ever planned on putting an Alid in power. And yes, the Abbassids didn't end up being much better for the remaining imams than the Umayyads were. It's not so ironic when you realize that their "pro-Shi'a" stance was never a true reflection of their ideals, just a way to get more people to join them.

For further reading:

The End of the Jihâd State: The Reign of Hishām ibn ʻAbd al-Malik and the Collapse of the Umayyads by Khalid Blankinship

The Prophet and the Age of the Caliphates: The Islamic Near East from the 6th to the 11th Century by Hugh Kennedy

And, although dated, The 'Abbāsid Revolution by M. A. Shaban

Pardon me for asking but what does "SAWS" mean?