After reading lots of ideas, including:
But how to get it to work without requiring a wrapper class? I combined the ideas discussed above into the following two extension methods.
Interface Dependency Injection
If you like using interface DI, as in Bar.Bar(IFoo foo) then use this one:
        /// <summary>
        /// Used to register <see cref="IHostedService"/> class which defines an referenced <typeparamref name="TInterface"/> interface.
        /// </summary>
        /// <typeparam name="TInterface">The interface other components will use</typeparam>
        /// <typeparam name="TService">The actual <see cref="IHostedService"/> service.</typeparam>
        /// <param name="services"></param>
        public static void AddHostedApiService<TInterface, TService>(this IServiceCollection services)
            where TInterface : class
            where TService : class, IHostedService, TInterface
        {
            services.AddSingleton<TInterface, TService>();
            services.AddSingleton<IHostedService>(p => (TService) p.GetService<TInterface>());
        }
Usage:
        public void ConfigureServices(IServiceCollection services)
        {
            services.AddHostedApiService<ITaskScheduler, BackgroundTaskScheduler>();
        }
Concrete class Dependency Injection
If you like to use concrete class injection, as in Bar.Bar(Foo foo) then use: 
        /// <summary>
        /// Used to register <see cref="IHostedService"/> class which defines an interface but will reference the <typeparamref name="TService"/> directly.
        /// </summary>
        /// <typeparam name="TService">The actual <see cref="IHostedService"/> service.</typeparam>
        public static void AddHostedApiService<TService>(this IServiceCollection services)
            where TService : class, IHostedService
        {
            services.AddSingleton<TService>();
            services.AddSingleton<IHostedService>(p => p.GetService<TService>());
        }
Usage:
        public void ConfigureServices(IServiceCollection services)
        {
            services.AddHostedApiService<BackgroundTaskScheduler>();
        }
Enjoy!