Having the right to work in the US doesn't entitle you to a job. And there are plenty of reasons government institutions would want to disallow non-nationals a position. The whole point of citizenship is, after all, buying into the system; if you're not "in" your allegiances or potential recourse can always be argued to be limited. Dual-Citizens, such as myself, are in a similar boat for the same reasons (unless they opt to renounce foreign ties).
I don’t believe this job actually would be possible as the level of clearance requested is usually denied to dual-nationals (without renouncing, of course).
Citizenship is a pretty common requirement in information security in sensitive industries, to use the example I'm familiar with.
It says "US ONLY US Citizen," but that only excludes people who still swear allegiance to King Charles.