The main reason why people care about linear algebra is that it lets you solve linear systems of equations (and perform related operations, such as projections). A linear system of equations has an immediate correspondence with a matrix of coefficients, a right-hand side vector, and a solution vector. For this reason, it is very natural to first talk about matrices and vectors (they can be used to represent concretely a linear system of equations), and then introduce the concept of vector space in cases where the abstract view can be clarifying or help with understanding.
From my perspective, the "right" way to teach linear algebra depends on the mathematical maturity of the students. If they are honors math majors, they can easily handle the definition of an abstract vector space right away. If they have less mathematical maturity, the abstract viewpoint isn't helpful for them (at least not without first familiarizing themselves with the more concrete concepts). Think about it this way: we don't teach school children about natural numbers and arithmetic by first listing the Peano axioms.
I don't have first hand experience of the French system, but from what I understand the approach there is more along the lines I'm thinking of, and the relative over-representation of French graduates among my more mathematical colleagues suggests it may be rather effective in practice.
I hesitate to call anything pedagogically "wrong" as people think and learn in different ways, but I think the coyness some teachers display about the vector space concept hampers and delays a lot of students' understanding.
Edit: Actually, I think the "start with 'concrete' lists of numbers and move to 'abstract' vector spaces" approach is misguided as it is based on the idea that the vector space is an abstraction of the lists of numbers, which I think is wrong.
The vector space and the lists of numbers are two equivalent, related abstractions of some underlying thing, eg. movements in Euclidean space, investment portfolios, pixel colours, etc. The difference is that one of the abstractions is more useful for performing numerical calculations and one better expresses the mathematical structure and properties of the entities under consideration. They're not different levels of abstraction but different abstractions with different uses.
I'd be inclined to introduce the one best suited to understanding first, or at least alongside the one used for computations. Otherwise students are just memorising algorithms without understanding, which isn't what maths education should be about, IMO. (The properties of those algorithms can of course be proved without the vector space concept, but such proofs are opaque and magical, often using determinants which are introduced with no better justification than that they allow these things to be proved.)