What makes us believe that a time-dependent metric is the right way to describe expanding universe instead of just claiming that galaxies fly away from each other?
I can see, that it allows them to recede with speed greater than the speed of light, but I am unsatisfied with only such a premise.
Can we say that they move in a different way than if they followed the geodesic in a static universe?
Historically, there was a cosmological model which adequately explained observations of galaxies flying away from us within a picture of unchanging at large scale universe: the Steady State Theory.
Within this model the universe is eternally and continuously expanding and so the existing matter is spread over ever increasing volume. In order to maintain the constant average density of matter there must be a process of matter creation, replenishing the matter that is being carried away by expansion. This model is illustated by a following space-time diagram (taken from [1]):
Note, the worldlines of galaxies have a beginning: they are created "from nothing", average density of galaxies remains constant, and observers in the central galaxy ("us") would see the that the more distant the galaxy the more redshifted it is (indicated by the angle of intersection of red past light cone with the worldline of the galaxy).
However, besides redshifted galaxies cosmological model must explain a lot of other observational data. In particular, steady state theory failed to adequately explain cosmic microwave background radiation (more specifically it high degree of isotropy and spectrum). As explained in [1]:
The Universe now is not producing a blackbody since it is not isothermal and it is transparent instead of opaque. In the Steady State the Universe was always the same so it never produced a blackbody. Hence the existence of a blackbody background ruled out the Steady State.
There were efforts (that to some degree continue now) to update the model to account for new observations with the Quasi-Steady State (QSS) cosmology. The discrepancies that still remain are summarized in [1].
Just to add to the previous person's answer. The reason we are interested in general relativity (GR) is phenomenological. Observations of things above planetary distance seem to line up perfectly with GR. At scales more than, say, 100,000 km, the cosmos seems to be as close to GR as we can discern. And in many ways those observations are exceptionally sensitive.
There are a few situations at short distance that we can get some tests of GR. But I am way out of date on those. Things like gravity wave detection. And there are some tantalizing situations where the results are very interesting. Like the dark matte problem. And certain star systems showing interesting behavior that seem like they are at the limits of accuracy. So far, GR is still the champion.
So, we think that the universe follows GR. And GR says, under a quite wide range of possible conditions that could result in what we observe now, that the universe should be expanding. And that means that the metric should be changing over time.
Another addition to the previous answer about what it means to be time dependent. The thing to do isn't to find coordinate free expressions. The thing is to look for scalar quantities that don't depend on coordinates. It's a subtle difference. The components of the metric will depend on the coordinate system you use. So, getting one component of the metric, in one particular coordinate system, isn't enough to say "time dependent" or "time independent." What you need to do is find a scalar quantity. As a motivational example, the wiki page on scalar curvature in Riemannian space is instructive.
https://en.wikipedia.org/wiki/Scalar_curvature
There is a little more complication in GR, because the manifold is pseudo-Riemannian, meaning it has 3 space and 1 time coordinates. But this gives the general idea. You construct values that do not depend on the coordinate system. An example might be a derivation of the Hubble constant in each phase of the evolution of a cosmology model. In the usual FLRW type model, the Hubble constant is not actually constant.
What makes us believe that a time dependent metric is the right way to describe expanding universe instead of just claiming that galaxies fly away from each other?
What you suggest saying "fly away from each other" fits to the Milne Model, which is a explosion scenario based on special relativity in flat Minkowski spacetime. The Milne model is mathematically equivalent to the 'empty FRW universe'. Its metric is also time dependent however. This is unavoidable if something changes over time. In other words the terms universe and spacetime are inseparable.
Can we say that they move in a different way than if they followed the geodesic in a static universe?
Yes, in Einstein's static universe timelike geodesics of comoving observers (who experience the universe isotropic) don't deviate from each other which contradicts ("fly away ..."). But also this metric contains dt² because peculiar velocities are always possible in any spacetime.
We have theoretical constraints from general relativity. If you make models in GR that are homogeneous and isotropic, you end up with a general class of models called the FLRW models.
You phrased your question in terms of a time-dependent metric, but this is technically not quite the right way to describe it. In GR, you can take flat spacetime (Minkowski space) described in the usual coordinates, subject it to a change of coordinates, and come up with a new description in which the metric varies. So instead of talking about whether an FLRW metric is time-dependent, we need to talk about time-dependence using definitions that are coordinate-independent. The coordinate-independent way of talking about it is whether or not the spacetime is static.
FLRW models are not static except in the special case where the density of matter is zero. We observe that the density of matter is not zero. (As pointed out by HeisenbergImage in a comment, you can also have the Einstein static universe, which is unstable.)
You phrased the question in terms of a time-dependent metric, which is not really a good way to define the issue. Depending on how you fix that problem, you could end up talking about the empty FLRW, Einstein static universe, the steady-state model (A.V.S.'s answer), or the vacuum dominated FLRW. None of these is an accurate description of our present or past universe, although the vacuum-dominated FLRW will eventually be a very good approximation.
