The first result on the google search "zeta function of grassmannian" seems to contain quite a direct and not too long derivation of the zeta function for a grassmannian over a finite field:
http://www.math.mcgill.ca/goren/SeminarOnCohomology/GrassmannVarieties%20.pdf
From the zeta you see that it is rational, of course get the zeros (which are none), but you don't immediately get confirmation of the functional equation. Though, from the very simple combinatoric representation of the zeta function, it might be easy to prove directly, I will try with pen and paper later.
I'm glad I searched this, I didn't know the zeta was so simple in this case as well
I would guess that Grothendieck's envisaged proof, via the standard conjectures, would be "morally right" in Kontsevich's sense. (Although there is the question of how the standard
conjectures would be proved; since they remain conjectures, this question is open for now!)
The objection to Deligne's proof is that it relies on various techniques (passing to symmetric powers and Rankin--Selberg inspired ideas, analytic arguments related to the positivity of the coefficients of the zeta-function, and other such things) that don't seem to be naturally related to the question at hand. I believe that Grothendieck had a similar objection to
Deligne's argment.
As a number-theorist, I think Deligne's proof is fantastic. One of the appeals (at least to me) of number theory is that none of the proofs are "morally right" in Kontsevich's sense. Obviously, this is a very personal feeling.
(Of course, a proof of the standard conjectures --- any proof, to my mind --- would also be fantastic!)
[Edit, for clarification; this is purely an aside, though:] Some arguments in number theory, for example the primitive root theorem discussed in the comments, are pure algebra when viewed appropriately, and here there are very natural and direct arguments. (For example,
in the case of primitive roots, there is basic field theory combined with Hensel's lemma/Newton approximation; this style of argument extends, in some form, to the very general setting of complete local rings.) When I wrote that none of the proof in number theory are "morally right", I had in mind largely the proofs in modern algebraic number theory, such as the modularity of elliptic curves, Serre's conjecture, Sato--Tate,
and so on. The proofs use (almost) everything under the sun, and follow no dogma. Tate wrote of abelian class field theory that "it is true because it could not be otherwise"
(if I remember the quote correctly), which I took to mean (given the context) that the proofs in the end are unenlightening as to the real reason it is true; they are simply logically correct proofs. This seems to be even more the case with the proofs of results in non-abelian class field theory such as those mentioned above. Despite this, I personally find the arguments wonderful; it is one of the appeals of the subject for me.
Best Answer
See Theorem 5.6 in Kleiman's article "The Standard Conjectures", in the Motives volume (PSPM 55.1). I'm not quite sure what you mean by "written completely in the language of motives", so it might not be exactly what you are looking for.
I initially posted this as a comment, but it really should be an answer, so I'm re-posting it as one.