Having both is strictly better then having one. But because classes need to be balanced, strictly better is not strictly better.
Existing support-contributing classes have both, have tremendous flexibility in choosing which to use, and have further flexibility to do blasting, rituals, single-target control, battlefield control, and a host of other things, including shapechanging and turning undead.
Balancing the Warlord is going to be a challenge, but not because there's any need to take things away from it to balance it with other support casters. The challenge is, how do you make a class that was so focused on "Leader" support abilities, which now constitute only a fraction of what other former-Leader-role classes now do in 5e, balance with those classes? More battle-field control through tactics? Some single-target control through 'hectoring?' Some improved personal offense via maneuvers? There's a lot of design space out there that the existing martial classes haven't touched, and it's not like 5e has been shy about classes overlapping eachother, either...
There seems to be a lot of people saying that healing and mitigation are equivalent. I say they aren't, and that's a great niche to put the warlord. Healing is retroactive, mitigation doesn't have to be.
I don't want "oh look, it's the same as the cleric".
The Druid and Bard are both full castes who fill the support role with some of the exact same spells as the Cleric. So, we certainly don't need more of that!
However, just by virtue of not casting spells or having supernatural abilities, at all, the Warlord is already more distinct from the Cleric class than any other support-contributing class or sub-class, bar none.
I'm not pushing for Warlord personally, but it looks like there is a lot of interest and I'm not one to say "badwrongfun, only my way is correct!". So let's make a cool warlord.
That is a fine & noble attitude. Hard to live up to, sometimes, but a good ideal to keep in mind.
And if people are asking for opinions (as a poll does), I would suggest instead of making them a healer clone to give them their own niche that is demonstratively mechanically different. Not a drop-in-replacement for a healer, but something that could play nice with a healer in the party if you have one. Maybe go with something like song of rest for out-of-combat healing, and action (proactive) or reaction based mitigation instead of later healing.
You're not going to make a class 'cool' by taking most of it's toys away. While those might be nice additions to differentiate the Warlord, it's hp restoration is already necessarily going to be different from the Cure Wounds and similar spells shared all the existing support characters. Triggering HD, preferable with a bonus to help the party get through the day, is perhaps one of the most often-mentioned possibilities for Inspiring Word.