Races would be definitely more distinctive with feats like that, but wouldn't "cool" actions like this push even more players choosing race for powers and not for their story? I truly am more of a fan of "roleplaying" feats that connect story with gameplay in that regard.
Humans were still the most popular race in 4e. Their "power" was getting to choose an extra at-will and an extra skill--and they were the only race that only got +2 to one stat, rather than two, albeit whatever stat you wanted. Objectively, most other races were at least a little more powerful starting out, and humans only really became
great as a pick after multiple years of getting more support than pretty much anything else.
An elven fighter should not feel that differently than a dwarven fighter in gameplay, but they should feel different connection to the world and story and context. If you choose an elve not because they can teleport, but because they can ask there great-great-great-great-great-grandfather for help who still lives in the undying court. (Just a maybe not so great example I thought in the moment)
Rules cannot realistically represent that element--so they should not waste effort on things they cannot achieve. That's (part of*) why 4e left stuff like that, on the level of "this specific character's roleplay," as something to be talked out between player and DM.
What rules
can do, however, is provide support for unusual expressions or combinations of ideas. As one example, 4e Eladrin (think "high elf, but its own distinct race") had interesting and distinctive support for the Fighter, Warlord, and Wizard classes (amongst others, those are just the ones I know relatively well.) Eladrin Fighters could benefit much more from their Intelligence score, which was normally not all that useful for 4e Fighters. Eladrin made
excellent "Tactical" Warlords (effectively, a subclass focused on speed and group maneuvering), and got feats and Paragon Paths that enhanced them further in that direction. And then as a Wizard, Eladrin could take the feat Eladrin Sword Wizardry, which would let them treat longswords as if they were wands, gaining all the associated benefits of using a wand, while still having a melee weapon equipped for when that was relevant. (This, also, paired very nicely with certain nice Wizard and/or Eladrin PPs.)
IMNSHO,
that is how you make different races feel different: reward interesting, flavorful choices that reinforce the theme and concept of that race. Build on a foundation set by the basic mechanics, and embellish those mechanics with support and options down the line. Anyone can be a Wizard--and, with a feat or two, even a regular Wizard can use a longsword as their spellcasting implement (="focus" in 5e terms). But only an Eladrin can use a longsword as if it were a wand.
When paired with the 13th Age style of stat boosts (one of two from race, one of two from class, your choices must differ), you get a system that is flexible enough to enable a consistent baseline of mechanical effectiveness, while still preserving the
feeling that playing an X is different from playing a Y, even with the same class--because the things you do and care about actually do differ. The equipment you use, the abilities you take, the actions you favor, they actually differ, because there's mechanical incentive to make them differ.
*The other reason is that the fanbase reacted with
violent anger at the built-in flavor of things like "Golden Wyvern Adept"--only to then rip the resulting books,
which did what players had asked for, up one side and down the other for not having enough flavor. D&D truly has an unpleasable fanbase sometimes.