Or did he mean that a cleric is required *if* you wanted characters to regain hit points FASTER than through normal overnight recovery?
I think I've finally realized the source of perhaps one of the disconnects.
Adopting 5e's terminology of the "standard adventuring day" as a shorthand for "the number of challenges and fights (measured in terms of XP) that a party should be able to handle before it needs to rest":
1. The presence or absence of a cleric should not change the length of an adventuring day. A cleric that prepares a healing spell could restore damage, but if that cleric was replaced by a damage-dealing ("striker") character or had prepared an offensive spell instead, he could have killed an enemy before it dealt that damage. If that cleric was replaced by a protecting ("defender") character or had prepared a defensive spell instead, he could have prevented that damage from being dealt in the first place.
2. Adding damage mitiation or avoidance abilities to all classes increases the length of the adventuring day. However, this would not have any effect on the "experience" of playing a cleric. The fact that a player is running a cleric who has prepared healing spells should be enough to ensure that the healing will be necessary, because the cleric is neither killing the opponents fast enough or preventing damage to the party.
3. One of the key effects of having a cleric would be to reduce the downtime between adventuring days. This is not an issue if each adventure consists of only one adventuring day, and the PCs have sufficient downtime between adventures to recover fully. However, if each adventure consists of more than one adventuring day (so that most parties will have to rest to recover resources at least once during the adventure) and natural healing is slow (because hit points are considered "meat") then magical healing is the only way to reduce downtime between adventuring days. If the party want to guarantee a source of magical healing, then someone has to play the cleric. Mind you, this happens only because of the traditiona disconnect between recovery of spell/ability resources and hit point resources, but that is a whole separate issue.
With that in mind, my answers to Dire Human's questions are:
1. Should the Core require Cleric healing? Is it OK to have a party of four non-healers survive an adventure long enough to get loot and go home?
Yes, if the adventure is one adventuring day in length. If the adventure is two or more adventuring days in length, then the length of downtime between adventuring days for a party of non-healers may become an issue.
2. Should the Core require Cleric healing? Is it OK to have different classes (Warlords, Bards, etc) that also fulfill the required healing role?
As others have mentioned, the Basic game only requires
cleric healing.
3. Should the Core require Cleric healing? Is it OK to have different types of abilities that work similar mathematically to healing (temporary hit points, parrying, dodging, etc) and one of those be required?
It's okay to have such abilities, and adding them would extend the adventuring day. However, none of these would actually make a cleric obsolete since during the adventure, the presence of a cleric with healing spells means that the healing spells will need to be used to get to the end of the standard adventuring day, and between adventuring days, none of these abilities actually work to reduce the downtime.