It's fine unless you're level 18
To see how Purple Dragon Knight (PDK) stacks up let's compare it to a fighter archetype from the PHB. Since we're worried about PDK being underpowered we only need to find one already accepted archtype that it is comparable to. We're going to use Champion because it is broadly similar and it's abilities make an for easy evaluation. Also we're only going to look at direct ability vs ability since the, "feel", of an archtype in play is too subjective.
Level 3
- Champion: Improved Critical for some occasional (1/20) bonus damage.
- PDK: A (fighter level) group heal once/short rest as a bonus action.
It's a judgement call but I'd say they are about equivalent at this level. Improved critical may or may not come up much based purely on luck but some extra group healing is always handy.
Level 7
- Champion: Half proficiency with some skills they may already have. Longer jumps.
- PDK: Extra skill proficiency. Double proficiency with persuasion.
About equivalent. Both archetypes just pick up some skills.
Level 10
- Champion: Another fighting style
- PDK: When you action surge one of your allies makes an extra attack. Goes up to two attacks from allies at level 18.
Since the fighting styles don't offer much synergy the extra fighting style is likely to give a bonus to the fighter's backup weapon or just +1 AC. A once per short rest extra attack for your allies is likely to do more damage in an average day; near certain to after level 18.
Level 15
- Champion: Improved Critical 2, the Critening.
- PDK: Bonus saves for allies a couple times per long rest.
Champion might be better at this level but it's a judgement call. If you value straight damage another 1/20 chance of critical hits is something. If you value survival clutch extra saves for your team is great. I'd call it a toss up.
Level 18
- Champion: Quick healing
- PDK: Your level 10 ability now works on two allies
This is the tipping point where Champion is clearly better than PDK. PDK's big finisher is just... a slight improvement of an earlier ability. That's pretty sad.
Conclusion
Below level 18 PDK is happily balanced versus Champion. At and after 18 PDK is just too badly hampered by its lack of an archtype-capstone ability. Luckily for PDK most campaigns are played well below 18 making this less likely to be a problem.
Note about the level of Inspiring Surge
While the book says the Inspiring Surge upgrades at level 17 that is apparently in error. This tweet from Jeremy Crawford says that it should be 18th level.
Those ethnicities are specific to Faerûn; other settings have their own ethnicities.
As you correctly surmise, the Player's Handbook clearly defines the nine major ethnic groups as specific to Faerûn, that is to say the Forgotten Realms, while the Sword Coast Adventurer's Guide contains further information specific to that same setting, given that the Sword Coast is a place specific to Faerûn. The Imaskari, for example, are descended from the people of the empire of Imaskar.
While officially no one setting is the "core" setting for D&D 5th edition, in practice, the Forgotten Realms is given focus in the Player's Handbook, considering that its human ethnic groups are listed there.
However, other D&D settings have their own ethnicities.
For example, Greyhawk. The prevailing ethnic groups are the pale-skinned Suel, the golden-skinned Baklunish (who destroyed each other's empires in an ancient magical cataclysm), the Flan (natives to the Flanaess, the primary continent of the World of Greyhawk setting), the Oeridians (founders of the Great Kingdom), the dark-skinned Olman, and the river-dwelling Rhenee. None of the Forgotten Realms ethnicities appear in Greyhawk, although someone can canonically travel between those two worlds.
In Eberron, humans have varying skin, eye and hair colour, but the setting glosses over the exact details, and does not name specific ethnic subgroups. According to a comment by Eberron creator Keith Baker:
This is a case where canon Eberron simply doesn’t make an effort to accurately model demographics in our world. The premise is that your human character can look like what you want it to look like, and we aren’t concretely mapping skin color to region; essentially we are looking at HUMANITY as a “race” and cosmetic variation within humanity as a player choice. It’s not realistic, and within your campaign you can certainly decide to do otherwise, but it’s not something that will be defined in canon.
Dragonlance has its own ethnic groups detailed here including the Abanasinians, Arktos, Cobar, Ergothians, Horselords, Ice People, Istarians, Kazar, Kharolish, Lemishite, Nerakan, Nomadic Humans, Nomads of Khur, Nordmen, Schallsea Folk, Solamnics, Tarmak, Tarsian, Thenolite, Uigan, and Wemitowuk.
Best Answer
Wizards hasn't published a “campaign setting” book for the Forgotten Realms for 5th edition. (Yet?) There material scattered through several books — the Sword Coast Adventurer's Guide, plus all the hardcover adventures except Curse of Strahd — but no definitive campaign setting book like we've seen in past years. Information scattered across multiple sources like that, and being only a small fraction of what a campaign setting for the Realms normally covers, makes that an unsatisfactory source.
Until a proper campaign setting is published, there's only the old campaign setting guides to really go from, plus the bits and pieces of updates scattered across various 5e products. Ironically, the fan-maintained wiki is the most up-to-date source for the Realms after the Second Sundering (i.e., the D&D 5th edition era).
Are the SCAG and the adventures worth picking up just for the lore?
If you're looking for setting material, I'd say… probably not the adventures? That would be so much material to sift through for just a few bits of lore. What you'd get would be very detailed for some small parts of the Realms.
The SCAG is a better prospect, as it's closer to being a setting guide. Limited as its geographical scope is, it may be worth it if you intend to adventure mainly on the Sword Coast. (And the Sword Coast is thousands of miles long, so that's not as limited as it could be.) I'd say you should leaf through it if you get a chance to see a physical copy — or look for reviews of SCAG that give you a feel for what it covers and how deeply — and make a decision then.