Lisp is so powerful, but without static types you can't even do basic stuff like overloading, and have to invent a way to even check the type(for custom types) so you can branch on type.
Which the inferencer probably can't do because of how dynamic standard-class can be. Also, if we want to get pedantic, method-dispatch does not dispatch on types in Common Lisp, but rather via EQL or the class of the argument. Since sub-classes can be made (and methods added or even removed) after a method invocation is compiled, there is no feasible way in a typical lisp implementation[1] to do compile-time dispatch of any type that is a subtype of standard-object.
Now none of this prevents you from extending lisp in such a way that lets you freeze the method dispatch (see e.g. https://github.com/alex-gutev/static-dispatch), but "a modern compiler will jmp past the type checks" is false for all of the CLOS implementations I'm familiar with.
1: SICL is a research implementation that has first-class global environments. If you save the global-environment of a method invocation, you can re-compile the invocation whenever a method is defined (or removed) and get static-dispatch. There's a paper somewhere (probably on metamodular?) that discusses this possibility.
Missed your reply when it happened. Note that you cannot specialize a method on "fixnum" since it's not a class (though you can on "number" or "integer" since they are classes).
