[RPG] How should a neutral character focused on balance be presented

I think the key issue here is that you're thinking of it as "the old system with alignments removed" rather than "the old system simplified." Character archetypes haven't gone away, there's just more variety within each alignment.

Your examples are pretty easy:

• Personified forces of nature (chaotic neutral), aka Greek mythology, which, while dangerous, are not necessarily evil?

  Neither particularly good, nor particularly evil? Sounds unaligned to me.

• The honorable lawful evil villains, who offer their opponent a fair chance in a duel?

  A villain, not driven by Chaos for the sake of chaos? Probably evil, although perhaps drifting into unaligned (that Lawful Evil/Lawful Neutral border has always been a soft one).

• What about the structure of the Planescape universe?

  If you wanted to port these over whole-cloth, you'd need to separate character-sheet alignment from planar alignment. Characters from a Chaotic Good Planescape plane would have a character-sheet alignment of good, but would have personalities different from characters from a Neutral Good Planescape plane.

• The endless war of the Tanar'ri and Baatezu?

  Two groups of people locked in a conflict of ideals. One beholden to the principle of Law, the other Chaos. Both evil. From an alignment perspective they'd be evil and chaotic evil.

Consequences

The long-and-short of it is that the mechanics of roleplaying in D&D have been simplified and softened. The guidance to the DM is to award experience for good roleplaying, but "good roleplaying" is no longer as well defined.

Some of the results of this are:

• Players who create a character, and then assign an alignment to it will have more freedom to create deeper, more nuanced characters, because they have more room to add interesting contradictions within each alignment.

• Players who pick an alignment, and then create a character around it will be more homogeneous, because there are fewer starting points to work from.

• The DM has fewer "sticks" with which to punish players for loose roleplaying. There was an article not long ago that talked about how power had slowly been shifting from the DM to the players in recent editions of D&D, and this is likely part of that. (If someone can psychically deduce what article I'm thinking of, I'd love to link it here).

Opportunities

Moral dilemmas where there is no obvious good and evil choice?

I'm not quite sure how the alignment system impacts this. This seems to be the sort of thing that's always been squarely in the DM's court as a writer. Certainly, the good alignment is enough to get players into plenty of conundrums on its own.

Possibilities for intrigue and conflict when a lawful neutral inquisitor-type character has to cooperate with a chaotic good one, to defeat a great evil.

Those possibilities still exist, the ball is just further in the players' court. The inquisitor might be lawful good or even unaligned, while the other character is simply good, but well-characterized characters will still find reasons to butt heads.

Of course, it still takes good roleplayers to make sure this sort of conflict doesn't go sour!

Both of your examples have been implemented time and again by DMs in systems outside of D&D. The alignment system, while interesting, has proven to be non-vital to the process of roleplaying.

“Always” alignment does not actually mean always

Always: The creature is born with the indicated alignment. The creature may have a hereditary predisposition to the alignment or come from a plane that predetermines it. It is possible for individuals to change alignment, but such individuals are either unique or rare exceptions.

Note that creatures with acquired templates do not experience “birth,” so the first line does not apply. Becoming a vampire “always” changes one’s alignment, but in rare cases (perhaps as in the case of your NPC), that can be subverted.

And even in the cases of Evil vampires, they can be reformed. There’s even an explicit spell for doing that (santify the wicked from Book of Exalted Deeds), though I strongly encourage you to completely ignore it as it’s very poorly designed (like most of that book), and has some extremely unfortunate implications (if you ignore the fact that the books says it’s good, and read it, it sounds like a pretty awful, evil thing to do to a person).

Thus, yes, you can have a non-Evil vampire. Redeeming Evil creatures is not just a Good act, it is the quintessential Good act.

That said, no character is ever required to always act for the maximum Good; it is not an Evil act to choose to not perform a possible Good action.

That said, assault and murder are pretty much definitively Evil...

Evil Alignment is consistently not an acceptable reason to attack someone

Attacking someone without specific cause is assault, which is Evil and in most jurisdictions illegal. Continuing that assault until the target dies is murder, which is definitely Evil and illegal most everywhere.

A paladin who attacks someone purely on the basis of pinging for detect evil should, under the rules, fall on the spot, for willingly commiting an Evil act.¹

This is described in multiple rulebooks. It’s one of the few things about alignment that actually is somewhat consistent.

Alignment is not a detailed or consistent system

Alignment is described in different ways in different books, and the definitions are vague, ambiguous, and conflicting. The system is a historical artifact of D&D’s roots: it is designed for a simplistic, hack-and-slash dungeon crawl, where the players are Good because they are the players, the goblins, orcs, and vampires are Evil because they’re the enemies, and no one ever thinks too hard about that. Unfortunately, D&D has evolved but alignment hasn’t evolved with it; though people play far more serious and varied games than a straight dungeon crawl, alignment is still the same nine boxes. Don’t expect much from it; I actually strongly encourage you to ignore it. Outside of those simple dungeon crawls, it causes more headaches and arguments than it will ever be worth.

¹ I cannot more strongly recommend against the actual falling rules, however. Instead of stripping the paladin of class features (boring, interrupts the story, punishes the player), I strongly recommend switching the character to the appropriate alternate alignment variant paladin, so he keeps his powers they just become “dark” (or chaotic if that’s the way he falls).