Hope it is not too late, I did some digging on my own and did not find enough evidence to set dynamic javascript methods using the set() method. Even returning concrete references did not help in this case. So here is my theory
When we use
iconID.set('v.onclick',component.getReference('c.showDefaultValue'));
We get the following error

As it states "Value provider does not implement set(key,value)", It looks like the set property is not available for the onclick attributes. Maybe it is working as designed. Salesforce does not want us to implement it this way.
Solution :
In our scenario, as you suggested earlier you could use an if-else condition to perform your logic. But it looks messy, you would have to go through 2 methods (openDropdown & showDefaultValue ) every time some-one clicks, and this would add complexity to the logic. Hence I propose the following one.
Use Dynamic Components instead of setting the values. It works kinda similar to the set() approach that you used.
Component :
<div aura:id="container" class="container">
<lightning:buttonIcon aura:id="downIcon"
iconName="utility:down"
variant="bare"
alternativeText="Settings"
iconClass="dark"
class="slds-button slds-input__icon slds-text-color--default"
onclick="{!c.openDropdown}"/>
</div>
Controller :
({
openDropdown : function(component, event, helper) {
console.log("**** Inside openDropdown method ****");
$A.createComponent(
"lightning:buttonIcon",
{
"aura:id": "closeIcon",
"iconName": "utility:close",
"variant": "bare",
"alternativeText": "Close me",
"iconClass" : "dark",
"class" : "slds-button slds-input__icon slds-text-color--default",
"onclick": component.getReference("c.showDefaultValue")
},
function(newButton){
var divComponent = component.find("container");
divComponent.set("v.body",newButton);
}
);
},
showDefaultValue : function(component,event,helper){
console.log("**** Inside showDefaultValue method ****");
}
})
Results :
Lightning button before clicking

Lightning button after clicking, it enters the opendropdown method

Finally, when clicking on the close button, it enters showDefaultValue methd without entering the opendropdown method. This approach kind of works in the same as your component.set() logic but in addition, it creates a dynamic component and sets the v.body attribute of the outer container to the newly created component.

Did you embed this in an object app builder page or previewing through dev console.
If you are previewing through dev console force events are available only inside a one.app container.
your app is a standalone and the resulting page will be directed to lightning.force.com your force events will not work as expected.
Preview from dev console redirect me to :
https://namespace-dev-ed.lightning.force.com/
As per official docs
This event is handled by the one.app container. It’s supported in
Lightning Experience, Salesforce1, and Lightning communities.
Best Answer
Wait, never mind! I've got it working. There were two things I hadn't done: