In documentaries the Big Bang is often represented as blackness followed by a big explosion with loud noises. This always feels cheap and falling short of explaining the origin of the Universe. Is this really the best way to get green-thumbs thinking about the origin of the Universe? Are there other, better, representations or analogies of the Big Bang? If so, how do they better describe the current cosmological model than the "explosion" analogy?
No, it's not the best reproduction of explaining the big bang. Generally when you think logically, it was an explosion of matter, however it was not a fireworks show.
To best explain it simply, it is conceived as a singularity that rapidly expanded. An important thing to understand is that within the beginning of the expansion, the matter was quark–gluon plasma and opaque, meaning there was no 'BANG'. Intensity of the heat was too great to allow quarks to form hadrons. Hadrons comprise of baryons such as protons and neutrons, at the earliest stage of inflation protons had not formed, being a representation of the 'big bang' they aren't implicitly going to introduce these factors. A general representation if the bang factor itself, if the bang is presented as an inflating balloon filled with gas, it's not going to attract. To be a true analogue, the balloon would simply appear into existence (between 10^-6 sec and 1 sec), already expanded and have pockets of matter now cool enough to begin producing light, within the next 10 seconds electrons begin to form attracting photons and neutrons, the first atomic bonds (hydrogen) would continue for the next 380,000 years. Much of the earlier matter has been annihilated by antimatter and we begin to see matter in the form it is today.
In a nutshell, it's hard to present this theory when something just 'becomes', so I believe they use the explosion as a way of demonstrating the nature of forces rather than an analogue of mere inflation.