Short Answer

Implement IApplicationFeatureProvider<ControllerFeature>.

Question and Answer

Does anyone know what “service” interface is responsible for [discovering all available controllers]?

The ControllerFeatureProvider is responsible for that.

And does anyone know of a way to make ASP.NET Core “dump” the names of all the controllers it discovered?

Do that within ControllerFeatureProvider.IsController(TypeInfo typeInfo).

Example

MyControllerFeatureProvider.cs

using System;
using System.Linq;
using System.Reflection;
using Microsoft.AspNetCore.Mvc.Controllers;

namespace CustomControllerNames 
{
    public class MyControllerFeatureProvider : ControllerFeatureProvider 
    {
        protected override bool IsController(TypeInfo typeInfo)
        {
            var isController = base.IsController(typeInfo);

            if (!isController)
            {
                string[] validEndings = new[] { "Foobar", "Controller`1" };

                isController = validEndings.Any(x => 
                    typeInfo.Name.EndsWith(x, StringComparison.OrdinalIgnoreCase));
            }

            Console.WriteLine($"{typeInfo.Name} IsController: {isController}.");

            return isController;
        }
    }
}

Register it during startup.

public void ConfigureServices(IServiceCollection services)
{
    services
        .AddMvcCore()
        .ConfigureApplicationPartManager(manager => 
        {
            manager.FeatureProviders.Add(new MyControllerFeatureProvider());
        });
}

Here is some example output.

MyControllerFeatureProvider IsController: False.
OrdersFoobar IsController: True.
OrdersFoobarController`1 IsController: True.
Program IsController: False.
<>c__DisplayClass0_0 IsController: False.
<>c IsController: False.

And here is a demo on GitHub. Best of luck.

Edit – Adding Versions

.NET Version

> dnvm install "1.0.0-rc2-20221" -runtime coreclr -architecture x64 -os win -unstable

NuGet.Config

<?xml version="1.0" encoding="utf-8"?>
<configuration>
  <packageSources>
    <clear/>
    <add key="AspNetCore" 
         value="https://www.myget.org/F/aspnetvnext/api/v3/index.json" />  
  </packageSources>
</configuration>

.NET CLI

> dotnet --info
.NET Command Line Tools (1.0.0-rc2-002429)

Product Information:
 Version:     1.0.0-rc2-002429
 Commit Sha:  612088cfa8

Runtime Environment:
 OS Name:     Windows
 OS Version:  10.0.10586
 OS Platform: Windows
 RID:         win10-x64

Restore, Build, and Run

> dotnet restore
> dotnet build
> dotnet run

Edit – Notes on RC1 vs RC2

This might not be possible is RC1, because DefaultControllerTypeProvider.IsController() is marked as internal.