In an OO language, the equivalent to simple pattern matching would be to dispatch on an abstract method of a shared supertype. If you add a new type and forget to implement the abstract method, you get a compiler error. There's your exhaustiveness checking and it's entirely modular as you don't have to touch your original code.
As I said, that's one side of the expression problem. OO languages conversely have a problem (unless they support multiple dispatch or methods external to a class) when they have to add new actions. With ADTs, that's just a new function. With inheritance, you have to add a new method to every single class that implements a variant.
There isn't usually a 100% modular solution, because you want to stay flexible respectively in either dimension, to some extend. You have to chose between trade-off (as always).
That's way I like hybrid languages like Scala where I can actually make an informed choice whether it's more likely to add new variants or more functionality in the future. But when I have to do the thing I didn't anticipate in the first place I want help from the compiler to avoid errors. That's way an exhaustiveness checker for pattern matching is a very welcomed and neat feature.
There are actually language mechanisms that can get you both. multimethods, for example, or their ultimate generalization, predicate dispatch. Few languages support them, though, because they are overkill for most application domains (though there are some, such as computer algebra, where they can be very useful).
Note also that while many OO languages constrain you in having methods only occur within a class definition, this is not a requirement. Multimethods invariably need to be more flexible (because there's no privileged "self" object), but the multimethod approach can also be applied to single dispatch. See also partial classes and related techniques.
Also, ADTs in their basic form are still a very specialized, limited form of sum types. Even if you want sum types, better extensibility is often needed (though ADTs remain a very useful special case), as well as the ability to actually describe subtypes of a sum type.
See how OCaml ended up with having polymorphic variants and extensible variant types and Scala with case classes, for example.
2. Multiple dispatch and static typing are completely orthogonal features.
3. Strictly speaking, you don't even need multiple dispatch. The key feature of multimethods that you need is the ability to add a method outside the class body.