What about the big bang model do you find attractive, and what might cause you to choose to "lack belief" in it?
First I have no faith to lose in the BB, as I don't "believe it", but rather accept the basic premise of the BB as given by observation.
Second, I think there is some confusion about what the BB model encompasses.
The BB model is a response to the observation that the universe is expanding (via the larger recession/redshift for more distant objects) and therefore must have been more compact, dense, and hotter in the past. Absent any source of "matter generation" the logical conclusion of the model is that the Universe was once extremely dense and hot, and perhaps existed as a "singularity" (though known physics breaks down before such a compression.)
Because of the high densities and temperatures we can predict from the BB model a few things:
1) That the universe was once hot enough that all atoms were ionized and therefore opaque. This means that at high enough redshift (far enough away) there is a "wall" of plasma through which we can't observe more distant photons. This is known as the cosmic microwave background (CMB) since its redshift makes its appear as a 2.7 K blackbody, rather than the several thousand degrees it was.
The BB model predicts "a" CMB, but not the specifics since that would depend on the details of the cosmic expansion, matter contents, and density. The measurement of the observed 2.7K blackbody tells us (or constrains) some of those parameters.
2) That the initial elemental content of the universe was set by the cooling of the expanding universe. The projection backward of the BB expansion goes to temperature where nuclei are cannot be bound (and eventually, even baryons [neutrons and protons] cannot be exist). Known as Big Bang Nucleosynthesis (BBN), the assembly of simple nuclei from neutrons and protons is predicted by the BB model. The precise nature of it (exactly how much of each isotope is produced) is regulated by the properties of the expanding Universe (notably the baryon/photon ratio) and the details of nuclear physics. The exact ratio of H to He (for example) is not a general prediction of the BB model, but a property of our Universe and its specific properties. Currently, BBN fits well with a single value of the baryon/photon ratio, with a little "tension" remaining that may be the result of incomplete knowledge of nuclear physics, or the need to include interactions with other particles like neutrinos, some of which are only theoretical at this point.
3) That the planets, stars, and galaxies formed from the primordial gas of the cooling universe. The BB model does not specify the details of large-scale structure formation, galaxy formation, or star formation, but only lays down the basic conditions from which such theories must begin.
The BB model :
1) DOES NOT require or prohibit a phase of rapid inflation in the history of the universe;
2) DOES NOT require or prohibit a phase of slower acceleration (dark energy) later in the history of the universe;
3) DOES NOT specify the matter content of the Universe (baryon content, dark matter, etc.);
4) DOES NOT specify the nature of the laws of physics (Standard model, etc.), but the things it generates must follow those laws;
5) DOES NOT specify the cause or origin of the Universe, or whether their are other universes.
The currently favored cosmology (LambdaCDM) is a BB cosmology, but adds to the basic picture by including an early inflation phase, a specific baryon/photon ratio during BBN, "cold" non-baryon dark matter, and cosmological constant/vacuum energy "dark energy".
To the currently favored cosmology, evidence for the lack of dark energy or that it is some sort of consequence of the nature of space-time and any number of things would cause me to no longer accept the current model, likely in favor of a model that better explained things (particularly things we don't know yet).
As to the BB model itself, it is well established and supported just like the basic aspects of biological evolution theory and I consider it highly unlikely that it would fall from favor given the evidence behind it.