I don't know if I see the benefit to an app service. It seems like with controllers and domain services you can get the same result. Can someone post a scenario when it would make sense to use an app service over controller/domain service?
Host independence - if your domain model is only exercised through HTTP calls, your controllers may be fine to keep complexity low but being able to call them from multiple hosts (think console application for an event queue, serverless, tests) can be beneficial. I am a fan of adding complexity as it is needed but unfortunately, a lot of developers will just copy the pattern that has come before. Especially if the initial plans have been lost to attrition.
Tests - mentioned above and pretty much just one side of the above cube but having the Application Service as a seam for writing tests against is often quite useful if you don't want to tie your tests to your host. Having said that, tools for testing ASPNET (an I am sure other technologies) in proc have come a long way over the years.
Reveal intent - Controllers, unfortunately, suffer from functionality gravitation. All functionality tends to be pulled into them. Are they accepting HTTP requests? Deserializing those requests? Converting to a command? To a model? Orchestrating the domain model calls for creating the model, domain services, repositories? What really is it's responsibility? The term service is soooo overloaded that teams I have worked with call Application Services Use-cases and name them for exactly what they are trying to exercise the domain to do.
Although you can of course use controllers as this entry point, you lose some flexibility but gain some initial simplicity. This is the exact balancing act you are playing when delaying the use of DDD adoption in the first place instead of standard MVC app with no strategy for managing complexity. Maybe if you are not seeing a benefit, the application has not reached the complexity needed to use DDD in the first place? It does come with a complexity cost.
With regard to domain service, they are really part of your domain and do the work whereas the application service is really the entry point that orchestrates the whole use case. Be careful of overusing domain services though, personally, I often view them as a failure on my part to find a decent model (but maybe that's just me).