Gases can only scatter light strongly if it matches a quantum transition (Rayleigh scattering doesn't involve quantum transitions but it's relatively weak). Quantum transitions can be rotational, vibrational or electronic (strictly speaking rotational and vibrational transitions are usually combined). Rotational/vibrational transitions have an energy that is generally in the IR range (which is why CO$_2$ scatters IR light) and electronic transitions have energies in the UV range (which is why ozone scatters UV). So there's a gap in the visible range. You say:
I also notice other planets in our neighbourhood weren't quite so lucky
but assuming you're talking about Venus, Jupiter, Titan etc, the scattering is from particles not from gases. After all, Earth's atmosphere isn't particularly transparent on a cloudy day.
It is a co-incidence that the Sun's light peaks in the visible region. Planets have been found around all sorts of stars, and if the star is cooler or hotter than the Sun it's spectrum will peak at a different wavelength. However it may not be a co-incidence that we evolved on a planet whose starlight does peak in the visible wavelengths. After all, if it didn't, life on Earth would probably be different and we probably would be here.
White sunlight is a mixture of a huge range of frequencies (i.e. colours), of which most - but not all - are in the range that are visible to the eye. White is not one single colour, it is a mix of many colours.
The eye has 3 different colour receptors (cone cells). Each type is most sensitive to a certain colour range. Their peak wavelengths are in the red, green and blue regions of the visual spectrum, respectively. The brain then mixes the signals from all three types, to produce the sensation of a "colour".
Because our eyes evolved to see in sunlight, if the red, blue and green levels of the perceived spectrum are approximately those of sunlight, we see that colour as being the same as that of sunlight - something we call "white". Again, white is a mix of colours.
There are other perceived colours that do not appear in the spectrum. A common example is brown. Like white, brown is a mixture of pure colours. No part of the electromagnetic spectrum is brown, but our brain interprets certain mixtures of wavelengths that way. Tree trunks reflect only those parts of the sunlight that make the brain think "brown".
In the same way, if something reflects all wavelengths of sunlight, the brain thinks "white". In artificial lighting, if we mix the appropriate amounts of red, blue and green light, all three types of cone cells get the right stimulation to make the brain perceive it like sunlight, i.e. white. If the mixture is not exactly right, the brain will see it as some other shade, like pink or grey.
In fact, different "white" lights have different colour shades. It is said they have a different "colour temperature". The temperature of the sun is close to 6000K and light of that colour temperature looks like sunlight. Many white LEDs have colour temperatures of around 9000K, and their light will have a bluish tinge. Incandescents globes are closer to 4000K and look reddish - but we call all of these lightsources "white".
Best Answer
Water is not transparent for deepUV and infrared. From the evolutionary point of view our eye developed to see electromagnetic radiation present at earth in the past (and now) - deep UV and infrared are absorbed by water vapor and other gasses in atmosphere - so there were nothing to see at these wavelengths.
Here is a nice explanation on why some things are transparent and some are not : http://en.wikipedia.org/wiki/Transparency_and_translucency#Transparency_in_insulators
Basically water is dielectric - and majority of pure dielectrics are transparent.
http://en.wikipedia.org/wiki/Electromagnetic_absorption_by_water