Having looked at the different classes, I thought it would be instructive to find out how the classes fare in terms of how accessible they are for the different races. Each race (except for humans) gets +2 to two different ability scores. It's mainly how that breaks down in terms of whether those races have an edge in terms of picking a class that caters to either one or both of their racial traits.
Thus I have them separated into three categories: Best (both ability scores are used for the class), 2nd Best (one ability score is a primary ability the class), and Not-Useful-At-All (where the racial ability score bonuses do jackshit). The only real things that stopped me from applying a character race to a class was if it made next to no sense to do so, hence why warforged got boned since I don't feel like they really feel right for the primal classes.
After parsing these, it's interesting about the results. The longtooth shifter has the most number of Best Classes at 5(Strength and Wisdom apparently being VERY useful), followed up by deva at 4 (Intelligence and Wisdom being handy as well). Many races had 3 or 2 Best Classes, but warforged, eladrin and shadar-kai have the dubious joy of only having 1 good Class for them. They are better off than humans, kobolds and gnolls, who have no ideal classes. Humans only get one +2 ability score (which they get to allocate), so they CAN'T fit the profile of having two ideal bonuses, but kobolds and gnolls have Constitution and Dexterity as their bonus ability scores, a combination that doesn't apply to ANY class.
For 2nd Best, humans kick ass since they can get a bonus for any class they want to essentially, having all 17 classes as good "2nd best" classes. Longtooth shifters, dragonborn, genasi, and warforged all have 6 classes that are 2nd best, Strength apparently being handy in that regard. There are a bunch of races with 5, 4, and 3 2nd best classes, but bugbears, half-elves and hobgoblins only have 2 each, Str/Dex (bugbear) and Con/Cha (half-elf & hobgoblin) apparently not giving them good options for classes. They are however more versatile than the githyanki (Con/Int) who only have one 2nd best class, that being wizard.
If I count the number of Best classes as twice as handy as 2nd best classes and I use a weighted average, here are the numbers in terms of versatility of races in D&D.
Human 5.667
Longtooth 5.333
Dragonborn 4
Elf 3.667
Githzerai 3.667
Goliath 3.667
Minotaur 3.667
Orc 3.667
Razorclaw 3.667
Bladeling 3.667
Deva 3.667
Genasi 3.333
Dwarf 3.333
Half-orc 3.333
Warforged 2.667
Drow 2.667
Bugbear 2.667
Doppelganger 2.333
Gnome 2.333
Goblin 2.333
Halfling 2.333
Tiefling 2.333
Half-elf 2
Hobgoblin 2
Eladrin 1.667
Shadar-kai 1.667
Githyanki 1.667
Kobold 1.333
Gnoll 1
Yeah, humans get all their joy from versatility and those peskey "2nd Best" classes. Gnolls: Not a common race for empire building.