As a person who only very lightly played 4e, what is the intense draw of the Warlord?
Others have covered some of it, so this may seem a bit repetitious now, but it was all of the following:
- Re-introducing the "leader of men" archetype. AIUI, before 2e, Fighters became legit landed nobility, with retainers, servants, and the responsibilities and rewards of being "lord/lady of the manor." 2e and 3e let that archetype lapse (until very late 3.5e with the Marshal.) Giving that archetype its own defined space was a popular move.
- Offering a "cerebral" martial character. This means both letting players play martial characters who valued mental stats as much as (if not more than) physical stats, but also giving players martial characters which emphasized tactical choices, deep and meaningful gameplay, and a party-wide perspective as opposed to a personal-focused perspective (which was an overall goal of 4e; the Warlord was just the most prominent demonstration of that goal.)
- Making a support character exciting, even daring. Support-focused characters are generally not very entertaining, but there are a lot of players out there who genuinely enjoy supporting other people, rather than being the ones doing the direct actions. The Warlord was the best example in 4e of a support-focused character that was genuinely a lot of fun to play. Part of this stems from the Warlord being a very active support, whereas D&D has traditionally focused on passive and reactive support. Another part comes from 4e actually leveraging the Standard vs Minor action distinction to enable Leaders, like Warlords, to simultaneously do something impressive (usually Standard actions) and do something to help out or patch up an ally (usually Minor actions.)
- Diversifying the martial class-space. Have you ever noticed how we have three or four different variations on "arcane full-caster"? Each with half a dozen or more subtle variations? Yet we have basically one "guy who fights with weapons." Warlord created diversity by emphasizing different aspects of the martial class-space: instead of Fighter having to be all things to all people when it comes to martial classes, it could be some things for all people, and Warlord (along with other classes) could fill other needs.
- It was "new" in the way Dragonborn were "new." That is, technically, something like it had existed in late-3e, the Marshal class. But having it front-and-center, PHB1? That was different, and it caught attention. This also meant that, because it had been around from the beginning, it got a lot of pretty good support, and became very diverse toward the end of 4e's run (e.g. the "lazylord" became one possible build path, though I personally favored Bravura, the "high-risk, high-reward" leadership style, kinda-sorta the "subclass" in 5e terms.)
Due to the combination of these effects, Warlords are iconic for 4e. They were quite popular while 4e was running, and their exclusion from 5e was a very noticeable...
choice, to use a more neutral term than I normally would.
In that light, how can you be a "Warlord" at 1st level...
Seems like something that should be a 3E Prestige class.
How can you be a "Paladin" at 1st level? How can you be a "Warlock"? How can you be a
Wizard, someone who has to have completed
years of education and training before they could attain such a lofty rank! How can a mere
novitiate call upon the great miracles of the gods?!
The argument is specious. We allow for varying levels of "background training" or the like before a character actively appears in the narrative. And where we choose not to do that, we write our own explanation of what the class is--potentially even making an outright break with the implied flavor of the name. "Warlord," "Marshal," "Captain," they're labels, just as "Barbarian" is a label which does not literally mean "a person who does not speak Greek, so they just babble 'bar bar bar.'"
Plus, just try proposing the addition of a PrC.
Any PrC. Doesn't matter how you spin it. You'll get shouted down by folks who hated the 3e version (for good reason, 3e PrCs were
bad design) and thus respond equally negatively to any proposed new ones. I would know. I tried. Even as just a trial balloon, even as literally just "if the issues could be fixed, would you consider this?", folks were near-unanimous, on multiple forums, that PrCs are
bad bad bad and should never
ever EVER be allowed to exist in 5e.
In terms of combat things, is Paladin the closest in 5e?
Not really. The fact it uses spells is a huge issue, same with Bard (the other primary "well isn't it just an X?" option.)
The community term for the brute mechanical function of the Warlord, or any other similar sort of thing, is "full Cleric replacement." That is, the Warlord could do everything the baseline Cleric could do through its class actions: it could heal (though not as well as a Cleric could), it could grant saves (ditto), it could buff (generally better than a Cleric would, especially for offense), and it could grant attacks (one of the best at doing so.) Notably,
raise dead was a ritual in 4e, which meant anyone could learn it; Clerics could get it inherently, to having Ritual Caster as a bonus feat, but a Warlord could easily pick up Ritual Caster and would spend exactly the same money a Cleric would in order to learn it. (Incidentally, this meant a Wizard could actually be even better at resurrecting the dead than a Cleric, as they got some rituals
for free, unlike most other classes that get the Ritual Caster feat as a starting feature.)
The Warlord was an entirely non-spellcasting character. She might be doing things that
in our world would be outright supernatural, but
in D&D are pretty much par for the course (even in 5e, consider Fighter self-healing). She didn't "shout wounds closed"--but she could help her allies survive
shock long enough to actually recover, so long as they still had some gas left in the proverbial tank. (Which, believe it or not, is actually more similar to real-life healing than D&D usually is. Many deaths from injuries would have been preventable if the shock those injuries caused had been addressed fast enough. That's one part of why CPR and AEDs are so important; they can keep a person alive through the period of shock, allowing medical treatment to restore homeostasis, and thus saving a life.)