Assad wasn’t “keeping Sunni terrorists down” when he was letting Zarqawi and Krekar’s al Tawhid and Ansar al Islam transit and train in the country....
while simultaneously supporting the Shia Hezbollah and Imad Mugniyeh.
It was Syria that trained and planted Captain Yee as a US military chaplain and had him smuggling material out of Gitmo.
He was supporting anyone who would make trouble for the US in Iraq to forestall an invasion of Syria, a possibility that senior DoD officials kept muttering about (and I supported at the time, largely because I trusted them to make the right calls). But he was killing Sunni Arab Muslim Brotherhood types who opposed him, and many of those also oppose the US. Note that al Qaeda is a Muslim Brotherhood offshoot, as are Nusra and ISIS.
Note also that this idea that Assad had full control over the country is not a good assumption. The Hama rebellion in the 80's almost killed Assad's dad. The Syrian economy was in worse shape in 2003, probably because oil-rich Iraq's economy was in a shambles due to sanctions, and Syria probably relied a great deal on that trade. Meaning that dissent, while below the surface, was bubbling along, breaking out into outright revolt in 2011. Any serious suppression of Sunni Arab activities vis-a-vis supplying the Iraqi insurgency would have resulted in an even more rapid deterioration of his popularity which was never high in a Sunni Arab majority country. Any financial aid would have served to help him buy off dissent, i.e. the same function served by the $3b a year handed out to Jordan and Egypt.