To me, the most interesting design happens not when you say "let's make a class that does exactly what it did in past editions" but instead ask "what should a character based on this concept be doing?" And then try to do something unique with the design and mechanics of the class.
This I entirely agree with (and posted elsewhere on this thread). I'm down with the concept of the warlord in 5e, but a warlord designed around 5e ways of doing things, not trying to mechanically import from a different edition that had it's own set of design philosophies that didn't match.
All of that said, the below is just mechanic musings, not endorsing that any of these abilities should or shouldn't be on the list.
Grant bonuses (or advantage) to initiative checks.
This was one of my favourite passive warlord bonuses, as so few things muck with initiative.
Granting the warlord proficiency on initiative checks might be a simple tweak. Or half proficiency to the party.
A flat bonus isn't very useful though. It's just something you write down and then forget about. Despite coming up in every combat, it's not memorable.
Having an ability to allow characters to reroll initiative, making it more active, would be much more interesting, even if it only affected one person or was usable once a short rest. Ditto having an option allowing the warlord to shift a character's position in the initiative order. Fun and tactical.
The recent Ranger/Rogue UA had the rogue with an active ability that adjusted Init but not above yours. Giving Warlords some expectation of not the lowest Init (use better of DEX or INT?) and then give them a power to lift up a certain number of people (CHR mod, min 1) who can hear/see with lower Inits to right after the warlord. This way it's actively triggered and not a math bonus quickly forgotten. And it could be a big bonus - that caster rolled a 1 and really needs to go earlier and you can pull them up to right after you no matter what you got.
(BTW, if you make the bonus to Init better, you need to bring down the number of allies effected. Right now the two working together cap it to a meaningful number, and gives you reason to go "I'm going light armor dex warlord to have a better Init as a valid build choice.)
Give an ally a save against an ongoing effect (and the Warlord also attacks).
Useful and very much in the wheelhouse of the 4e warlord. But it feels like one of those abilities that better fits the "leader" role and not the "tactical commander" concept.
It's a checkbox ability that feels less vital to the class concept and probably more at home at a healing subclass.
It could be a simple as also allowing the Help action to benefit a saving throw instead of an attack roll.
To expand on your idea and actually cover a number of concepts that the warlord should be able to do: What if Warlord can offer help to any ally who can see and hear them as a bonus action. Help make saves? Done. Give advantage to their next attack? Done. Help on grappling or breaking out of web or other skill checks? Done. It's a not huge power in terms of lines of rules written, but it covers a lot of the concept. And it has precedent, such as the Master of Tactics from the Rogue (Mastermind). Also keeps everything on people's own turns.
(Alternative would be if an ally fails a d20 roll allowing a reroll with a reaction. But I'm not sure I like that as much. Difference between taking an doing something (setting up an ally to succeed) and reacting to something (helping an ally who is messing up). Plus it steps on the toes of bardic inspiration.)
Use an interrupt to reduce damage to an ally.
As simple as adding a "aid AC" to the Help action. A "Hinder" action if you will.
Protection Fighting Style and the Cleric (Protection) do this, can fit.
