I agree with your points in the post. The local cultures are the ones that matter to the gameplay. So when players play different core species, the DM needs to think about which cultures the characters are participating in. Example, have all of the citizens completely assimilated into a shared culture, or is the local culture a tapestry that still preserves some threads of cultural diversity, even if weaving together in a new way locally?
The e24 Players Handbook should and does avoid too much cultural lore. It resembles the 1e Players Handbook, with some one-liner suggestions for inspiration, and the setting that the DM uses determines everything. However, the e24 DMs Guide when officially offering the Greyhawk setting should help the DM with a bit more local lore (and background options) relating to the what Dragonborn are doing near Greyhawk City and more generally in the respective Flanaess regions. The hostilities of Tiamat as a main setting theme merit a comment about how various Dragonborn individuals and communities respond. I am satisfied with the setting implication that the Goliath originate from Jotens and are politically active in Yeomanry. But Dragonborn merit slightly more lore from locales.
The species and cultures dont need to be one on one. It is possible a local setting has two main cultures, such as a culture teaching arcane-primal values originating from Human, Gnome, Elf, Dragonborn, and Goliath, plus a culture teaching martial-divine values originating from Dwarf, Halfling, Tiefling, Awsimar, and Orc. Psionic might be known but considered private without public institutions. For the arcane culture I had Greyhawk setting Mage Valley in mind, with Giant and Dragon in surrounding mountains. For some recent historical reasons the two cultures are now citizens of a same place. The focus on cultures can do cool and interesting things.
Regarding the Elf species, I like how e24 formats them as a single species mechanically. 4e and e14 have been a process of consolidating the over hundred kinds of D&D Elf down to the mechanical stats of fewer species. Eventually, one species. The mechanics of Elf lineage represents an inherited transmission of magic. The family lines imply a family origin (Fey Crossing, Wilderness, Underdark, etcetera). The magic of each lineage exhibits an ancestor who adapted magically to an environment. Today the descendents may or may not live in the same environment, and a family of diverse ancestors may have siblings exhibiting different magical lineages. The mechanical format emphasizes that the Elf species is innately magical and is diverse. There are elven vestiges of mechanics from earlier editions in Mordenkainen Presents (Sea, Eladrin, and Shadar-kai) plus elsewhere Astral. But as DM I can easily use the e24 Elf format for them instead, by selecting the relevant spells and cantrip-like feature to represent their lineage. The Magic The Gathering Elves already do it this way.
Consider the Elf of the "High" lineage. It consolidated many different kinds of D&D Elves. Because they all now use the same stats mechanically, the entire diversity is now obviously different cultures, relating to background abilities and factions.
In the Greyhawk setting the High magic correlates with the following cultures: High, Grey, Faerie, Valley, and perhaps various other communities, such as Tulani Eladrin. These are defacto separate cultures.
Note, the 4e Eladrin being more urban magitech and less wilderness seasons, is the e24 High lineage. It is different from the seasonal e24 Eladrin lineage, and different from the diverse but hierarchical lineages of the 2e Celestial Eladrin.
So even in the single old school Greyhawk setting, the High elven magic alone exists in at least five separate native elven cultures. Add in the Wood magic of Wood and Grugach, and the fact that each Drow magic city has its own religious traditions and cultures, the one Elf species mechanics enjoys a virtually endless diversity of cultures, suitable for most setting needs.
The elven cultural diversity actually presents a similar lore challenge for the DM that Dragonborn do. Just because the character exhibits "High" magic, says nothing about which culture the Elf character grew up in. Is the character a Fey from the otherworld of Faerie? Are the parents fom a nonlolth Drow city in the Underdark. Do some ancestors come from Mage Valley? Is the family in Greyhawk City for generations? The family might even be Grugach, but the character is High magic because of some ancestor? It helps to have a richness of Elf lore to determine the identity of the individual character. And I would also like Dragonborn to enjoy rich lore options. But ultimately the player and the DM must decide together on whatever lore is going on locally.