No ordinary commercial flight will land at an airport in a disaster zone as you describe. However, it isn't impossible, and special crews trained in disaster relief operations do it all the time.
First of all, ATC is not required. All pilots are trained in landing at uncontrolled (aka pilot-controlled) fields, which comprise the vast majority of airports. The US alone has ~15,000 airports and only ~500 of them have a control tower. And most of those control towers close at night.
Secondly, ATC does not need radar, just radio sets which are easily flown in if all the ones on the ground are destroyed. The FAA even sets up temporary towers for air shows or fly-ins at normally uncontrolled fields. Likewise, the US military (and presumably others) practice landing at fields in hostile territory, securing them and setting up ATC operations. Radar improves safety, efficiency and capacity, especially in bad wether, but even most towered airports have no radar coverage.
Therefore, the actual humanitarian flights, even only a few days after the disaster hits, can have a relatively normal aviation support infrastructure available. It's the ground situation that keeps normal (scheduled commercial) flights away.
The obstacles are primarily legal and political. While many countries are quick to accept assistance, some are not. Getting permission for the flights themselves, customs waivers for the cargo and visas for the relief workers can sometimes take days or even weeks depending on political issues—and nobody is flying in until they're sure they won't be shot at, either by the local military or militia groups that may have taken advantage of the disaster. There may also be insufficient ground infrastructure to move supplies and workers from the airport to where they're actually needed as well.