They can do all they want when initiative isn't being tracked.
Up to whatever limit is placed on Inspiring Word or similar abilities... (Oooh, this'd be a neat way of dealing with the issue of 'abusing' at-will hp-restoration out of combat, and it'd make a certain amount of sense: in-combat Inspiring Word restores hps, out of combat, it grants temps for the next combat.... if you do away with HD or otherwise find the normal use of Inspiring Word unpalatable, you just change it to always temps.)
The only way HD-granted healing works is if it itself is not at-will (X per short rest) or if it has limitations on use (only triggers when the PC is below half-hp or at 0 hp).
HD /are/ a limitation on use. A per day (per 2 days, really - they're the most limited, slowest-recovering resource in the game).
The other option is to make warlord healing strictly inferior (1/2 roll or something) so that its used as a last resort vs. resting healing.
Strict inferiority is never an option. But being able to trigger a HD as an action or as part of an action or a bonus action would never be strictly inferior to waiting an hour to do so. I like the idea of maximizing HD spent on a short rest - it makes it a clearly more efficient option (if you have the time), and it speeds up boring bookkeeping, while making the use in-combat (rolled) more exciting.
HD are already pretty inadequate, and adding to or maximizing HD would be a way of extending the hp resources of the party to handle the full 6-8 encounter day, without actual outright healing resources like Cure Wounds.
That being said, 5e doesn't assume, as 4e did, that you get a short rest after every encounter. So its quite possible that you just don't get a chance to do this sort of advantageous healing ALL the time. 4e also had a similar loophole, the 'multiple short rest drill' where the cleric burned his remaining Healing Words, then rested, then burned them again, rested again, etc (each rest being nominally 5 minutes this was often not a huge issue plot-wise). It definitely maximized HS utilization, but if its expected behavior then its not a BIG issue.
The 3e WoCLW accomplished something similar, too. The big problem with extending hp resources via a WoCLW wasn't extending hp resources - all that does is let the party handle a longer 'day' without dying, and the main effect of that is greater pressure on other daily resources, like spells - it was taking the use-for-healing pressure off of CoDzilla's spell resources. In 4e using surges more efficiently just extended the day, but it didn't free up substantial spell resources (healing spells were generally utilities, or already bundled with an attack, so trading them out for something else was minimally disruptive to balance). In 5e, healing spells are taken out of the same slots as higher-spotlight-impact spells, so more efficient use of HD because of a Warlord could free up some healer spell slots, but the encounter and game balance are so much loser that 'disruption' may hardly be noticed - besides, a second caster would have a much bigger impact, anyway.
Of course if it seems to be TOO problematic then Inspiring Word can be limited to 'twice per short rest' or 'use a tactical die' or whatever the mechanic turns out to be. In 5e only a 10 or less CON character can even do 1x on average, and then requires 2 days to fully recover by the standard rule. Upping that to 1.5x every 2 days isn't such a huge difference that it is likely to break entire adventures and make warlords seem mandatory. That also assumes that MOST of the warlord's healing is done out of combat, which is ignoring its most advantageous use scenario.
Again, I think I'd have to see actual specific workups of the class to be able to criticize the mechanics at this level for sure.
Yep. It's fun to speculate about possible mechanics, but it's too early to draw conclusions from such speculations. To really get into it, we'd need something official, like a UA version of the class.
CW is limited by the number of cleric spell slots. Potions of healing either cost GP (50 per shot) or are rare and can't be made (without the DM allowing it). Both expend a resource that doesn't refresh quickly, making them valuable to share.
And a HD trigger is even more limited, because HD are the single most limited resource in the entire game, taking twice as long to refresh as spells, and representing only a modest amount of healing, in total, that can't be shared out to the character who needs it the most the way spells can be.
Both of your examples have an opportunity cost. What is the cost of spamming Warlord healing vs. taking a short rest? There is none. In fact, there is a reward (+Cha) to using the quicker method.
You're assuming one possible example of the mechanic in a vacuum. What if the bonus applies to short rests, too, or was better for short rests, for instance? What if it couldn't be spammed? It wouldn't be difficult to find a solution to the issue you bring up, the hard part would be deciding on which perfectly good solution to use.
Actually, that makes it worse. Since short rests are generally rarer, the ability to "top off" by using a HD without taking a short rest means PCs no longer have to gauge how hurt they are between combats (the trade off: do we wait and heal or press on) since they can effectively enter every combat again near full hp.
Which would reduce the number of short rests per day, which would hurt a Warlord with a significant proportion of short-rest-recharge abilities...
And pity poor bards; in an hour they can grant you an extra d6 of healing per HD.
And they're full casters. No pity available, sorry.

I knew there was already a way to make HD during a rest much more efficient. That's an average +3.5 /per die/ and you're whinging about a secondary-stat-mod to one or more dice? And, it's on top of being able to throw out true healing in combat via Cure Wounds or Healing Word, on top of that greatly increased HD efficiency.
My personal desire for warlord healing is to be situational.
'Healing' is situational by definition. It's of no use when no one's wounded, is only nice to have when someone is, but is critically important when they're dropped, for instance.
That's the problem, everyone is arguing in a void because nobody is agreeing on what the warlord's parameters are beyond "tactics", "buffing" and "healing".
And even 'healing' isn't exactly right, since the warlord Inspires allies to restore hps, he doesn't shout their wounds closed. But, yes, it is absolutely too soon to be getting into specifics. First we have to get WotC to get the class in the pipeline, a UA or something, then we can worry over the specific mechanics.