Concept for a Magic Melee DPS

[Archwizard]

I assume you're referring to the post I made about various ways to implement hybrids, which was written to convey some general ideas. If that's the case then you either misread or did not read the write up I linked you the last time we spoke. I suggest giving it another look (the URL is in my sig), as Dualcast is there and still proccing off spells. There's even a section in the write-up explaining how that interacts with the other mechanics I gave the job.

I'm referring to when you wrote such memorable lines as:

Assuming you want melee & magic mechanics, you can go the way of weaponskills granting RNG procs for instant spells, melee skills granting a stacking buff that reduces cast time of the next spell, or melee strikes building a resource that allows certain spells to be cast instantly. To ensure the rotation between sword and spells is followed, have melee strikes gain a secondary effect if used after a spell, or have spells apply a short debuff that increases damage taken from your melee skills.

It's a core melee job in Int gear (and even that would be a point of contention).

I'll agree with you here. My earlier RDM write-ups had it as a DEX DPS, though that's largely because a ton of DEX gear was given aesthetics that suit RDM (Amon's hat comes to mind). I'm willing to be flexible on this, though.

It doesn't matter if you can still proc Dualcast at range, since your core DPS rotation (not just the burst, but your main spammable combo) would be executed in melee and any casting benefits would be as "tacked on" as you claim melee is now. If you spend any significant period limited to melee, it's a melee DPS.
You've created Dragoon where Piercing Talon also gains a melee combo bonus (or High Jump and Mirage Dive are on the GCD), that doesn't make it any less token as a ranged attack if it's jammed between two melee attacks for it. Except in this case you've given the ranged attack(s) a cast time, so even with Dualcast (whose proc chance you've reduced, meaning even more time hard-casting at a distance), no melee would actually use it in any realistic scenario where they're forced to be at-range.

But of course,

On a personal note, I find the idea of hitting an enemy with a sword, followed by throwing a fireball at them at point blank range very satisfying.

... which still doesn't make you any more of a caster than RDM is a melee at present.

As for your writeup, I stress once again that it would be an amazing basis for a Spellblade/Mystic Knight/Rune Fencer, but you've so fundamentally changed it from the current RDM (you have about 7? of the same skills, most of those are in-name only and then rest are utilities non-essential to the rotation) that it's a completely different job to start with.
Imagine what you could do with that concept if you weren't fettered by having to call it a Red Mage -- branching combos with Enspells, cycling elemental and leeching Strikes, Warp Strike as a gap-closer, I had completely forgotten about Alisaie's Aetheric Blade (which would make an awesome limit break!). Slap on Maiming gear for a Knight look, and the actual support of melee role actions (of which you would get a lot more mileage than the caster ones), and...

I mean if we're being serious about designing Mystic Knight,

Kind of the thread theme here, mate.

the rules that pop in my mind are:

a) it's entirely melee-focused aside from the mandatory ranged skill like Piercing Talon and Knife Throw. Which means it won't be slinging spells and instead rely entirely on their magical sword(s) to deal damage
b) it won't offer utility along the lines of healing or rezzes.

a) Which is, by and large, what you've designed anyway. As I've said before, it doesn't matter if you have three Piercing Talons instead of just one, you're not going to be running in and out of range with the boss unless they force you to. Flinging a spell at point-blank range is little different from the act of any other point-blank melee attack -- again, NIN says hi, and SAM is happy to share its cast times.
b) We don't even use Vercure for utility, it's just a means to proc Dualcast during downtime or self-heal when soloing. The only times I've cast Vercure on another party member were when healers were stuck in animations for getting rezzed or casting LB, because it otherwise costs rDPS to use every time. So you're pretty much just arguing the rez on a Spellblade, which is getting a bit greedy.
At any rate, who says melee jobs can't offer that kind of utility? DNC may not be melee but it offers an oGCD heal, as does SMN, so it's at least an open possibility, and it wouldn't be a terrible stretch for a Spellblade to offer some kind of group mitigation instead of a heal. (And while I'm sure you were simply trying to limit it to "won't offer restoratives", on the off-chance you meant "utility in general" I would remind you that there is so much potential for En-spells.)
Besides, without the ability to get full mileage from Bloodbath, another self-sustain tool is a virtual necessity.

Put simply, I don't understand how giving the original hybrid of the Final Fantasy series more melee uptime can be so egregious.

In itself the request is innocuous, but it ignores the simple fact that in the current system for a Red Mage, even if melee time per combo (or should I say, Temps d'Escrime?) wasn't static, you'd still be limited by the fact that each melee skill is proportionate to a number of spells cast to generate it. Even our single cheapest Enchanted melee skill is roughly proportionate to two spells.

If it was purely about increasing melee uptime in the current system, you'd be focusing on improving the spells available to Red Mage in order to shift the Mana Economy and increase melee combos per minute.

The problem is that most requests for "more melee" on RDM don't think that far ahead, simply requesting a number of additional Enchanted melee skills to use per rotation, which would then extend the casting periods before we even get to the melee combo. It would surprise a lot of those folks when they spend just as much (if not more) time casting as a result because they've increased the cost of the rotation.

On the other hand, if it was purely about making RDM more like its classical iteration, we'd see something more like BLM where the goal is to burn out your MP as quickly as possible with Black and White spells, then swap to melee until you restore MP -- which would of course go directly against Red Magic being "more efficient" than Black or White. I'm fine with the current iteration.