Is it possible that the engineering mistake was to choose to build a city below sea level?
Nawww,,,,,, we can do that!
No. That was a political mistake. The engineering mistake was in determining the building and maintenance requirements (the first and fourth steps in most standard engineering projects).
When I was in the Navy operating nuclear reactors we had to periodically test everything to make sure that our equipment operated up to specifications. You never build something and expect it to last forever. In some cases it was difficult to test something by conventional means so alternate tests were performed (such as using radiography to test welds). But nothing was ignored.
Contrast this with the levees in New Orleans. How did they know (and certify) each year that the levees would operate to specifications? They didn't in reality. They just signed paperwork to make it look like they were. The most significant flaw was the settling of the levees and the sinking of the city so that the levees were overtopped. By measuring the height of the levees they could have determined if they needed to be repaired.
The biggest issue in this disaster is that the Army Corps of Engineers gave custody of the levees to the corrupt New Orleans Levee Districts assuming that since levee failure would hurt them the most, they would do a good job of making sure the levees were in specification. What a mistake!
Additional testing is more expensive. Perhaps New Orleans and the federal government wouldn't have bought the levees if they realized that. But that was a political issue. As far as I can tell, the initial design engineers did their jobs correctly. The maintenance engineers and politicians fouled this hole situation up.
No, zoning board mistake.
"Is it possible that the engineering mistake was to choose to build a city below sea level? "
The Dutch seem to do ok.