At first, I was able to get this working by extending the default RenderController used by Umbraco. That let me catch every umbraco page request and examine the requested content, looping through any block lists and searching for a certain element then populating its extended partial class properties. This felt really gross performance-wise and left all this element-specific code in one huge global intercepting controller.

A better solution, however, is to use .net ViewComponents. This lets each block list element have its own controller, that can populate data from an outside source. They become almost like mini hijacking controllers.

Interestingly, Paul Seal has another video about viewcomponents (https://www.youtube.com/watch?v=YcVzaXgPknQ) but I went a step further and invoked them from inside my block list elements.

In this way, the razor markup for my block list elements only contains one line, which invokes a viewcomponent, passing along its own Model to be the model of the viewcomponent

@await Component.InvokeAsync(nameof(ContactUsComponent), Model)

Then the viewcomponent 'controller' can populate or overwrite any properties on the model (I use a partial class to 'extend' the element view model) The constructor of the view component can be injected with any dependencies you need.

Its 'Invoke' method can take args, which I used to pass along the model

public IViewComponentResult Invoke(ContactUsElement contactUsElement) (
    contactUsElement.Hours = "GetMeFromAnApiCall";
    return View(this.GetType().Name, contactUsElement);

and the view component's view contains what would normally be found in the element's cshtml.