Racial features that don’t imply biology impacts culture
The racial feature changes given below are designed to erase all trace of any core setting from the player's handbook races. One or two of the PHB races have features that are much better suited to backgrounds. It makes no sense for an entire species of humanoid to know how to use a bow and arrow!
The rule for identifying these traits is simple: if it doesn’t make sense for a baby of that species to have the trait, it doesn’t make sense for the trait to be tied to race.
Hopefully, these changes will open up some new options for you. This Dwarven species can appear wherever you like in your homebrew setting, and it won’t have some curious, unexplained knack for hammers and smithing.
All Races
The following traits can replace the Languages trait
Languages. You learn two languages of your choice.
Dwarf
The following traits can replace the Dwarf’s Dwarven Combat Training, and their Tool Proficiency and Stonecunning.
Natural Brawn. When you make an attack roll with a melee weapon that you’re not proficient with, you add half your proficiency bonus to the roll, rounded down.
Stone Scent. Dwarves have evolved a specialised sense of smell that lets them detect mineral content. Whenever you make a Wisdom (Perception) check to identify the origins of minerals, metalwork, or stonework, you can add twice your proficiency bonus, instead of any proficiency bonus you normally apply.
The following trait can replace the Mountain Dwarf’s Dwarven Armor Training trait.
Tough Skin. While you aren’t wearing armor, your AC is equal to 13 + your Strength modifier.
Elf
The following trait can replace the Wood Elf or High Elf’s Elf Weapon Training or the Drow’s Drow Weapon Training.
Elven Legedermain. Your slender fingers make your movements especially swift.
Once on each of your turns, you can gain an additional object interaction. For example, you could draw and stow a weapon, or drink ale and open a door on the same turn. You have a number of uses of this trait equal to your Proficiency bonus, and you regain any expended uses when you finish a long rest.
Rock Gnome
The following traits can replace the Artificer's Lore and Tinker traits
Specialist Lore. A rock gnome’s mind is wired to meorise particular subjects obsessively. You gain an area of memorised expertise. Choose a specialist subject from Science, Nature, Arcana, Technology, or Culture. Whenever you make an Intelligence (History) check related to your chosen specialist subject, you can add twice your proficiency bonus, instead of any proficiency bonus you normally appIy.
Artificing. Rock gnomes are naturally touched with artificing magic. By performing a 10 minute ritual, you can enchant a tiny nonmagical object with one of the following magical effects. Choose one of the following options:
Clockwork Toy. When placed on the ground, the object moves 5 feet across the ground on each of your turns in a random direction. lt makes noises as appropriate to its appearance: for example, an apple will make crunch sounds as it rolls, or a tiny frog will croak.
Fire Starter. The object produces a miniature flame, which you can use to light a candle, torch, ar campfire. Lighting or snuffing this flame requires an action.
Music Box. When tapped, the object plays a single song at a moderate volume. The box stops playing when it reaches the song's end or when it is closed.
You can’t have more than three objects enchanted at a time. You can touch an enchanted object to end the enchantment on it.
Traits you may also wish to change
I think these traits are acceptable, as I think they are more tied to genetic makeup than cultural influence. If you disagree, drop me a line at @henrynicholass1 on twitter, or u/henry_smithy on reddit (currently inactive)
Halfling - Brave
Half-orc - Menacing
Art - Proud Dwarf by EdCid on Deviantart

Exotic Races
Githyanki
The following traits can replace the Martial Prodigy trait
Natural Precision. When you make an attack roll with a melee weapon that you’re not proficient with, you add half your proficiency bonus to the roll, rounded down.
Martial Defense. You are naturally limber and agile. While you’re not wearing armor, your AC is 13 + your Dexterity modifier, and your movement increases by 5 feet.
Hobgoblin
The following traits can replace the Martial training and saving face trait
Natural Brawn. When you make an attack roll with a melee weapon that you’re not proficient with, you add half your proficiency bonus to the roll, rounded down. Additionally, while you're not wearing armor, your Armor Class is equal to 12 + your Dexterity modifier.
Saving Face. If you fail a saving throw, you can gain a bonus to the roll equal to your Proficiency bonus. Once you use this trait, you can't use it again until you finish a short or long rest.
Kenku
The following trait can replace the Kenku training trait
Copycat Training. Kenku learn remarkably fast. By spending an hour with another creature that is proficient with a certain skill or tool, you can gain proficiency in that skill or tool.
You can’t have more than two proficiencies at once with this feature. If you already have two proficiencies from this feature, you can use your action to forget one of those proficiencies, allowing you to replace it with a new one.
You can’t gain expertise in a proficiency you have learned in this way.
Lizardfolk
The following traits can replace the Cunning Artisan and Hunter’s lore traits.
Cunning Precision. Your claws are specifically adapted to hunting, flensing, and crafting. When you attempt to craft a nonmagical item, you can use a corpse in place of any other required materials. Crafting an item in this way doesn't require you to expend gold. Once you use this trait, you must finish a long rest before you use it again.
Born to Hunt. Your strong eyesight, natural camouflage, and precise sense of smell make you a keen hunter. You gain proficiency with two of the following skills of your choice: Perception, Stealth, and Survival.