I've run into the exact same problems you have, but whenever I bring it up, people accuse me of being a min-maxing powergamer.
Anyway, here's my solution. It works with the core math of the game and allows you to use the points-buy system as written:
Generate ability scores as normal.
All racial +2s are now +1s. Races like mountain dwarf that get two +2s instead have an extra +1 into an ability score of the player's choice.
The character gains an additional +1 to one of three ability scores, depending on the starting class. This represents the tendecy of certain classes to appeal to individuals with higher scores in abilities.
So, for example, if I wanted to play a wood elf barbarian based on the Picts from the film The Eagle, and used the Standard Array for my ability scores, I would probably put a 15 into my Strength, a 14 into my Constitution, and a 13 into my Dexterity. As a wood elf, my Dexterity score would increase by +1 to 14, and my Wisdom score would increase as normal.
As a barbarian, I could increase my Dexterity by another +1 to 15 (essentially the same as the normal rules in the PH), increase my Str to 16, or my Constitution to 15.
The math remains balanced - only I've chosen a character for the unique, flavorful racial stuff instead of the boring math.
This method breaks down slightly with the variant human (by increasing the same stat with both the racial and class boosts and then taking certain feats, you can increase a score to 18 right off the bat), but can easily be mitigated by adding an additional rule to the variant human stating that they can't use a class ASI on the same ability score they used for a racial ASI.
