The below code worked perfectly fine for me

<aura:component implements="force:appHostable,flexipage:availableForAllPageTypes,flexipage:availableForRecordHome,force:hasRecordId,forceCommunity:availableForAllPageTypes,force:lightningQuickAction" access="global" >
 <aura:handler name="init" value="{!this}" action="{!c.doInit}"/>
 <lightning:inputRichText aura:id="body" />
</aura:component>

Controller code is as below

  ({
     doInit : function(component, event, helper) {
       component.find("body").set("v.value", "<p>This will be init properly once I switch to ContentNote</p>");
     }
 })

The below code worked perfectly fine for me

<aura:component implements="force:appHostable,flexipage:availableForAllPageTypes,flexipage:availableForRecordHome,force:hasRecordId,forceCommunity:availableForAllPageTypes,force:lightningQuickAction" access="global" >
 <aura:handler name="init" value="{!this}" action="{!c.doInit}"/>
 <lightning:inputRichText aura:id="body" />
</aura:component>

Controller code is as below

  ({
     doInit : function(component, event, helper) {
       component.find("body").set("v.value", "<p>This will be init properly once I switch to ContentNote</p>");
     }
 })