The Sword Coast Adventurer's Guide includes a new martial archetype for fighters, the Purple Knight of Cormyr. Comparing its traits to the fighter archetypes in the PHB, it seems really underpowered.
It looks like the Purple Dragon Knight (p. 128) is intended to be focused on the roleplaying "pillar" of gameplay, but I think I'm missing how it's at all comparable to the other martial archetypes.
I'm not heavily invested in the Forgotten Realms setting; are there enough setting-specific benefits to offset its mechanical weaknesses?
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.
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.
About equivalent. Both archetypes just pick up some skills.
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.
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.
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.
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.