I still don't know what playstyle you're going for. For example, a wizard goes through all their spell descriptions to find the correct one. The Barbarian recklessly attacks to bait enemies into attacking them and mitigating the damage. The Paladin attacks in melee and carefully pays attention for opportunities to use their rare resources to their maximum effect.
What's the swordmage's playstyle? And would it be unique enough to justify playing it?
Well, just out the gate, "unique enough to justify playing it" is rather in the eye of the beholder, so I'm a bit leery of the question simply from that. "Recklessly attacks to bait enemies into attacking them and mitigating the damage" might be adequate for some and utterly inadequate for others.
But, for my part, if you want a response in the same vein--focused on the mechanics rather than thematics--I'm taking a lot of inspiration from FFXIV's Ninja job. That is, in that game, Ninjas learn mudras, "hand seals" if you're familiar with the terms from Naruto (IRL, "mudra" refers to ritual gestures in Hindu, Jain, and Buddhist traditions); specifically you learn Ten ("heaven"), Chi ("earth"), and Jin ("man.") Using these mudras, you access various magical "ninjutsu" abilities; the number and sequence of mudras used before the Ninjutsu action determines which one happens. Using a single mudra always produces a thrown, fiery-looking shuriken ("Fuma Shuriken"). Two- and three-mudra combinations have a bunch of different effects, e.g. Ten->Chi or Jin->Chi produces a lightning bolt ("Raiton"), while Chi->Ten or Jin->Ten produces a short-range fireburst ("Katon"). This creates a playstyle where you have only a few discrete actions, but can potentially have a large number of effects, and chaining them together successfully is vital; if you use an invalid combination, you lose that opportunity to do anything (and instead get a small rabbit dancing on your head; this is a reference to past FF games).
So, the goal of the swordmage is to
build up to the best magic they can achieve with the attacks they have available. A Paladin is looking to exploit any particular successful attack, and expends their magic in the doing. A Swordmage is viewing the whole attack holistically and trying to assemble the best thing they can. So, for example, perhaps when they use Spell Combat they must declare a
sequence of runes before the attacks are rolled. To execute the spell they want, they must get enough successful attacks to actually cast it; if not, they only get a partial sequence and must make do with what they get. This leads to a much more precision-focused playstyle. The Paladin wants to hit
hard when she hits at all, but is fine with missing some so long as those hard hits happen. The Swordmage wants to hit
well, even if each individual hit is weaker, because success is dependent on hits of any kind.
This leans into the idea of a precise duelist (very appropriate, given how Swordmage is usually presented as being akin to an
épée or sabre fencer), and can be enhanced by adding thematically appropriate actions or incentives. Fencers usually have one hand free, hence why I suggested rewarding players with a free hand or who hold a spell focus (like a wand) in their off-hand. They also tend to exploit mobility as their best defense, which would be made into a serious contender against heavy armor by layering in some ability to teleport, levitate, or otherwise move across the battlefield in ways no other martial character does.
Shield is a staple spell of D&D, so perhaps that could be factored in as well, taking cues from the 3e PrC Abjurant Champion.
So: the Swordmage is a preternaturally mobile precision attacker who must build up a sequence of attacks in order to unleash the full potential of their magic.
As stated, I fully expect at least one person to scoff at this and call it inadequate to justify its existence, but it looks to me like something not well-served by any of the existing options. Neither Paladin nor Eldritch Knight
builds up their power, and both are mechanically and thematically encouraged to go for heavier defense and generally (though not
absolutely) do not receive particularly significant support for mobility or precision. If a Paladin could choose to miss twice as often but make every melee attack a crit, they almost certainly would choose to do so; a Swordmage would
never make that trade. If a Barbarian could always deal retaliation damage for times where enemies deal any amount of damage to them, they'd take it in a hearbeat, whereas a Swordmage would almost certainly refuse such a thing because they don't have the hardness or hardiness to withstand such attention. The Barbarian takes stupid risks because for them, the risk is no longer totally stupid. The Paladin hunts for hard hits that can be magically upgraded into telling blows. The Swordmage makes calculated assaults and prefers to wear enemies down by a thousand cuts...or, y'know, the lightning bolt that follows after that.
Others have mentioned the aegis and off-turn effect stuff; that would be an ideal inclusion if one can fit
everything, but I understand that what is ideal and what is practical do not always meet, particularly in the more limited mechanical space of 5e classes and the brevity of most 5e combats.
Well most Ranger spells actually don't have a problem with thrown weapons. Hunter's Mark, Ensnaring Strike, Hail of Thorns. Actually, Lightning Arrow does exactly what your Thor idea does. Hammer and all. You could even do it with a Hammer of Thunderbolts and get the ultimate "Bring me Thanos!" Experience.
Making something like this dependent on a specific magic item is pretty much right out. Waaaaaaay too many DMs seem to think that giving players a magic item they like is one of the most offensive things a player could desire. This is, however, a great idea for a one- or two-step lightning-based spell for a Swordmage. Perhaps the first rune is something like "Speed" and the second is "Lightning." Also, sorry, Ranger is simply flat not going to cut it with its massive emphasis on nature and survival stuff--that's far too baked into the core, and has nothing whatsoever to do with being a Swordmage other than a player specifically desiring that flavor.
Though I have to love how """natural""" the language used in 5e is here. Thrown weapons
are not ranged weapons, ever, regardless of their use-case; they are always melee weapons no matter how they are used. But when you use a thrown weapon to make an attack against a target
by throwing said weapon, that attack is a ranged weapon attack
but not an attack with a ranged weapon. Because of course it would be the former and not the latter, that's by far the most obvious and natural reading!