I thought so, because there is the use of api.onPageChange(url) already there a being executed just fine. Note that I am adapting the code from here, so it seems it’s very well true that I have access to the api object in the setupComponent method.
Just for further clarification, here is what I got so far:
With the code
<script type="text/discourse-plugin" version="0.8">
    const user = api.getCurrentUser();
    console.log("/u/" + user.username + "/billing/subscriptions");
    api.registerConnectorClass("above-user-profile", "back-button", {
    setupComponent(args, component) {
    api.onPageChange((url) => {
        if (url === "/u/" + user.username + "/billing/subscriptions" ){ 
          document.querySelector("html").classList.add("custom-homepage"); 
          component.set("displayCustomHomepage", true); 
        } else {  
          document.querySelector("html").classList.remove("custom-homepage"); 
          component.set("displayCustomHomepage", false); 
        }
    });
    }
});
I get the correct output /u/Miles/billing/subscriptions out of the console. But when I do,
<script type="text/discourse-plugin" version="0.8">
    
    api.registerConnectorClass("above-user-profile", "back-button", {
    setupComponent(args, component) {
    const user = api.getCurrentUser();
    console.log("/u/" + user.username + "/billing/subscriptions");
    api.onPageChange((url) => {
        if (url === "/u/" + user.username + "/billing/subscriptions" ){ 
          document.querySelector("html").classList.add("custom-homepage"); 
          component.set("displayCustomHomepage", true); 
        } else {  
          document.querySelector("html").classList.remove("custom-homepage"); 
          component.set("displayCustomHomepage", false); 
        }
    });
    }
});
there is no output to be found…