The context free languages are a strict superset of the regular languages, meaning that all regular languages are context free and there's at least one context-free language that isn't regular. As a result, any algorithm that can parse arbitrary CFLs can be used to recognize regular languages.
In the example you linked, you are absolutely right that the grammar provided describes a regular language and in principle if you were just trying to build a recognizer for it you could just use a DFA. Using Earley's algorithm has a few advantages, though. One big one is that, since the language is described as a CFG, you can extract a parse tree from the Earley parser when a string is in the language, meaning that you can recover the structure of the formula. This isn't something that DFAs can do by themselves.
As for whether Earley parsers can handle all CFLs and whether they're just extensions of DFAs, yes, Earley parsers can handle any CFL, and in some sense, yes, they're kind of extensions of DFAs. Context-free languages have an associated class of automaton called a pushdown automaton that is essentially a finite automaton hooked up to a parsing stack. Many context-free parsing algorithms, including LR parsers and Earley parsers, are essentially pushdown automata whose states control the stack.