Here is my take on it. I think that there is enough (could add a little more on some maybe) flavor to the various races outside of stat increases that differentiates them from each other. The stat increases represent a generalization of that race overall. Generalization is for NPC's, not PC's. PC's are all about going outside the norm for everything they can do. So a Goliath NPC represents his great strength by having that +2 STR baked into his NPC stat array which makes him "stronger" than the average (NPC) human/elf/dwarf/whatever.
So now tie stat increases to Class and Background. As has been suggested I would not tie a +2 or +1 stat increase to the save proficiencies of that class as that could hurt some builds. The DEX Fighter for example. Giving him a +2 STR does nothing for him. So for each class give, say, a +2 to one primary stat predetermined by class. Maybe Fighters get a choice of STR, DEX, or CON. Then they get a +1 to any other stat of their choice. Backgrounds would then give another +1 appropriate to the Background that cannot stack where the +2 from Class went. So you could get say +1 INT from Scholar but not +1 STR (but Worrgrendel, those books are HEAVY) but you could not put the +1 into INT if that is where you put your +2 from Class. I feel like this would open up more Race/Class combinations for most players that might not be explored otherwise because of the "I gotta max my primary stat or I'm behind the curve" thought process. I know that some people already don't min/max like that (we have a STR based Gnome Fighter in our current campaign) but I feel it does make for more flavorful options (Dwarf Wizard, Half Orc Bard, etc). I also feel like having one +2 and 2 floating +1's helps out those Classes and Subclasses that suffer from MAD.
What about inhuman Elven agility you say, like Legolas? Well, lets think about this for a minute. If you are making a badass Elf Archer like Legolas that skateboards down Olphant trunks and stairs you are maxing your DEX anyhow so why does it matter if the +2 came from Race or Class? But, say you are making an Elf that dumps DEX. Based on the Race giving +2 DEX and you were using point buy, that means your 8 is now a 10. How inhumanely agile is that? Average human is the answer so that +2 DEX is not really representing any kind of inhuman agility normally found in Elves. Plus, maybe I wanted to play an clumsy Elf? I guess he's still clumsy by Elf standards and that just means that Humans are being stereotyped by Elves as being "clumsy"? No thanks.