From a completely subjective point of view, the thrust of your argument makes intuitive sense--why should a method that is not patent eligible be able to skirt this result by reciting a general purpose machine for its execution?
Unfortunately, in the legal world, intuition must sometimes give way to legal formalisms.
In this case the formalisms are the statutory categories of invention. Processes (methods) constitute one distinct statutory category, and machines (apparatuses) another.
Getting back to your point, you are correct to say that, after Bilski, a method (process) claim that is otherwise not patent eligible may not become eligible by simply providing that the disclosed method is executed on a general purpose machine (computer).
HOWEVER, an apparatus claim is separate and distinct from a method claim that is tied to an apparatus.
So while it is true that tying a method claim to a general purpose computer is not sufficient to overcome Bilski's machine requirement, it remains true that the holding of Bilski DOES NOT APPLY TO APPARATUS CLAIMS.
As an example, consider the following:
Method claim: "A method of ... comprising the steps of ...."
Method claim tied to a general purpose computer: "A method of ... executed by a computer processor, comprising the steps of ...."
APPARATUS claim: "A computer readable storage medium containing instructions, when executed by a computer processor, operable to ...."