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.
Weil's proofs for curves and abelian varieties essentially use special cases of the standard conjectures, and the framework of the standard conjectures (like many other conjectures of a motivic nature) is suggested by trying to generalize the abelian variety case (or if you like,
the case of H^1) to general varieties (or if you like, to cohomology in higher degrees).
Deligne's proof for K3 surfaces uses a motivic relation between K3s and abelian varieties (which is most easily seen on the level of Hodge structures) to import the result for abelian varieties into the context of K3 surfaces. This is not so different in spirit to Manin's proof for unirational 3-folds, except that the relationship between the K3 and associated
abelian variety (the so-called Kuga-Satake variety) is not quite as transparent.
[Added, in light of the comments by Donu Arapura and Tony Scholl below:] In the K3 example, it would be better to write "a conjectural motivic relation ... (which can be observed rigorously on the level of Hodge structures) ...".
Best Answer
I guess that "just the Riemann hypothesis" means the statement about the eigenvalues of Frobenius acting on the $H^i$ of a smooth proper variety, and "all of then" means the full theory of weights : definition of mixed sheaves, how the 4 operations affect weights etc (from which you get the hard Lefschetz theorem, the decomposition theorem...).
So here is my understanding :
Deligne's first proof : just the Riemann hypothesis.
Deligne's second proof : the full package for $\ell$-adic complexes, using the same kind of ideas as his proof but improving them.
Laumon's proof with the $\ell$-adic Fourier transform : It allows to simplify parts of Deligne's second proof, but does not replace all of it. More precisely, Laumon gives a shorter proof of theorem 3.2.3 of Deligne's "La conjecture de Weil II" that doesn't use section 2 of that paper, but he still needs results from section 1 (the calculation of the monodromy of lisse sheaves on curves in 1.8 and the fact that "most" irreducible lisse sheaves are pure proved in 1.5.1). Once you have theorem 3.2.3, it is pretty easy to deduce the rest of the "full package" from it, so Laumon doesn't say anything about that.
Katz's proof : Katz gives a simpler proof of theorem 3.2.3 of Deligne's Weil II paper, or rather of a weakening of it that is enough to deduce the Riemann hypothesis and hard Lefschetz. It is not enough to deduce the full package immediately, although this can be done without too much pain. I am not very familiar with Katz's proof. It seems to be different from Deligne's and Laumon's proofs but to use the same kind of techniques as Deligne's proof. My impression is that it is more concrete that Deligne's proof. (Note : I am not convinced that the difference between "just the Riemann hypothesis" and "the full package" is so big. The reason is that, if you know the Riemann hypothesis, then you should be able to deduce the full package for complexes of geometric origin. But global Langlands tells us that, over a smooth curve over a finite field, every irreducible lisse sheaves is of geometric origin up to twisting by a rank 1 sheaf. So in a way everything is of geometric origin. The situation is very different over a number field, of course.)
Kedlaya's proof : It is supposed to be modeled on Deligne's second proof modulo the simplification of Laumon, but of course the actual technical are different because Kedlaya uses p-adic coefficients. Also, he got the Riemann hypothesis but not the full package, because at the time the theory of relative p-adic coefficients was not fully developed. But now it is !
Abe and Caro's proof : See http://arxiv.org/abs/1303.0662 They develop the theory of weights for overholonomic D-modules with Frobenius structure on varieties over finite fields and obtain a full Weil II package for them (and even the Asterisque 100 extension, ie the theory of weights for perverse sheaves). They say in the introduction that their method is independent from Kedlaya's method.