Upon closer inspection, such claims appear to actually state that
diamonds cost more than their abundance alone would account for.
Sources that outright claim that diamonds are common don't seem to
have anything to back it up with.
In terms of where diamonds can be found, this is true.
Note that I believe a comparison of diamond production to gold production to be intrinsically flawed. The geological processes which create the deposits of each are vastly different. Gold is actually extraterrestrial in origin, whereas diamonds form from materials already on our planet over billions of years. There is a vast difference in the density of the materials, and gold's value is based solely upon purity of the metal, whereas determining value of a diamond is much more complex, and varies wildly from specimen to specimen.
A much more useful comparison is to other gemstones.
Mindat.org lists 738 localities for diamonds. Compare this to the 20 for benitoite, a very rare gemstone, the 381 localities for ruby, the 284 localities for emerald, or even the 626 localities for sapphire.
Note, however, that locality alone is a poor indicator, as the overall productivity of a mine (amount of desired resource retrieved) can vary significantly, and, given that the full extent of a site's production cannot be measured until it is fully exhausted (and even then it is possible to miss materials).
In 2013, rough diamond production from mines was estimated at 130 million carats. By contrast, in 2005, emerald production was estimated at 5,400kg (27 million carats).
Gem-quality diamonds only account for approximately 25% - 30% of the production of diamond mines*, which would mean that the global production of gem quality diamonds is estimated at between 32.5 million carats and 39 million carats in 2013. By definition, emeralds are gem-quality specimens of the beryl mineral family with a rich, distinctly green color, so 100% of emerald production can be considered gem-quality.
This puts global gem-quality diamond production at roughly 20% greater than emerald production.
Top-quality emeralds, sapphires, and rubies are rarer than top-quality diamonds, and can command a higher price-per-carat, but generally, other qualities of those gemstones are lower than comparable prices for diamonds.
It is difficult to evaluate comparative value vs comparative rarity when discussing gemstones, because the value of individual samples of gemstones varies tremendously based upon the specific characteristics of the specimen. In particular, color, size, and clarity (the amount of, or complete lack of, other mineral inclusions) impact the value of a given gemstone, although other factors such as cut, fluorescence, etc., can also increase or decrease the value.
*note that the proportion of "gem-quality" diamonds seems to be the topic of some debate. Between 3 sources (1, 2, 3), the proportion is respectively 20%, 25%, and 30%.