(This post isn't so much tied to the current train of discussion in this topic as just something that's been churning around in my head for a fair bit, and this seems like as good a place as any to share it.)
A lot of the traits that species currently get are more cultural than anything biological or magical: most proficiencies - weapon, armor, or skill - most spellcasting on non-planetouched species...ideally, I'd personally want to move those traits over to backgrounds, so that the species-based differences are things that are actually innate - size, senses, natural weapons/armor, damage resistances, condition immunity, natural magic, that sort of thing.
A number of playable D&D species, particularly animalfolk and planetouched, already have inherent traits that make them mechanically distinct from other species (or would if they were made stronger, like with various types of natural weaponry or natural armor). Most others have traits that could be leaned into more to make them mechanically distinct: Dwarves have evolved to live underground, Elves are fey, Goliaths are giantfolk that are adapted to mountain life, etc. In a paradigm without species-based ability modifiers, and where cultural stuff isn't inescapably entwined with innate stuff, there would be more space to lean on those existing hooks to give those species traits that make them feel different in play - less of a risk of "humans in masks" sentiment when each species is doing something fairly unique to it.
(For Humans, I'd lean somewhat on the "natural versatility" thing - codify Prodigy as an innate Human mechanic, though likely spread out over a few levels: skill proficiency of your choice at level 1, tool proficiency of your choice at level 3, expertise of your choice at level 5. Then I'd either give them an ability that lets them heal more per Hit Die spent on short rests, or maybe just an extra Hit Die or two, which covers both our rapid healing and our relatively high endurance - we bounce back fast, and we can keep going after most other species run out of gas.)
Orcs are among a relative few playable D&D species without a fairly obvious hook to build around: the others being other "monstrous" species (goblinoids and Kobolds) and the +2 Int species (Gnomes and Vedalken). Of those, Kobolds and Goblins are the only ones I've had much luck coming up with that hook for:
Kobolds sort of fit in the space of being to Dragonborn what Halflings are to Humans, as a small species adapted to life among a similar, larger sentient species, so a mixture of "little" mechanical traits and "draconic" mechanical traits produces something that feels appropriately Kobold...but there's still the question of how to make them feel like their own thing, rather than a Halfling/Dragonborn hybrid.
Goblins are significantly shorter-lived than most other playable species: where Elves and Dwarves can differ from Humans in mindset due to their extended lifespan, Goblins go in the other direction, being even more focused on the here-and-now, on making the time they have count. (There was a podcast I listen to that leans into this by, in part, having its Goblins basically sprinting everywhere: they're either already where they need to be, or getting where they need to be as quickly as possible. They don't really do half-measures.) The issue there is how to convert that mindset into mechanical stuff: maybe Goblins can spend their Hit Dice to get some sort of immediate benefit, or some other "glass cannon" ability? Maybe give them an extra reaction, to show how they cram more doing stuff into their time? Or just give them Dash and/or Use Item as bonus actions rather than Hide? I don't know, but there's at least something to build around to make Goblins feel distinct. I'm not sure I could say the same about the other goblinoids...
(Vedalken vary enough from plane to plane that the only consistent thing about them is their color, both in terms of skin and of magic. But blue skin doesn't translate to anything mechanical in D&D, and being creatures associated with blue mana doesn't distinguish them in D&D, where the colors of magic don't exist, or at least don't have any mechanical impact. As for Gnomes, I've never understood what their niche is: Dwarves are tinkerers, Elves can do the "forest friend" stuff, Halflings already cover the "little folk" trope...why are Gnomes a thing?)
While I haven't come up with an answer to the question in this thread's title, I do think this frame of reference narrows it down somewhat: what is Orcs' hook? What's their ecological and mechanical niche? If you can come up with a solid answer for that, you'll have something that marks them as distinctly Orcish, no matter what culture they were raised in or what god or gods may or may not have sway over them. Relentless Endurance seems like the best place to start, as the most iconic Half-Orc trait, and also the one trait among Half-Orcs or Orcs that skews the most biological (and, arguably, the farthest away from the problematic tropes that are at the root of the desire for change in the first place): they have a limited ability to just refuse to die. There's got to be something to work with there, right?
