Why was it that people had no trouble believing Saddam had chemical weapons?

Oh yeah.
In March 1988, the town and the surrounding district were unmercifully attacked with bombs, artillery fire, and chemicals. The chemical weapons included mustard gas and the nerve agents sarin, tabun, and VX. At least 5,000 people died immediately as a result of the chemical attack and it is estimated that 7,000-10,000 were injured.

This video was shot by the first reporters to enter Halabja after the attacks.
(Warning on content)

Chemical Ali was executed today in Iraq. He was notorious in the 1980s and 1990s for his role in the Iraqi government’s campaigns against internal opposition forces, namely the ethnic Kurdish rebels of the north, and the Shia religious dissidents of the south. He was responsible for the mass killings of Kurds at Halabja.