Like I said, I'll need to see it in action. It's possible that everything, taken together, will balance out in such a way that the limited magical healing available will actually make a meaningful difference, enough to counter-balance the offensive disparity between a Cleric and a Barbarian.
I doubt healing spells are meant to balance the Cleric in that sense. If anything, the 'healing burden' helps balance all the other stuff Clerics get relative to other casters (heavy armor, divine domains, channel divinity, d8 HD, decent weapons, etc) on top of getting all spells known automatically and having just as many slots, because healing eats into those slots to benefit other characters.
I'm not terribly hopeful, though. It sounds like they haven't left much room for error. Prescribed schedules rarely hold in reality, (especially for any type of sandbox DM). It certainly seems like it will be another 4E situation, where everyone can heal themselves between encounters, and the Cleric might only be useful to bring people up from zero in the middle of combat.
I agree we can't say how it'll play out with regard to pacing incentives, atm. While I can see how HD & overnight healing are going to help encourage the party to take a short rest each day, and not wait too long between long rests, I'm not sure I see what (beyond DM intervention or player restraint) is going to prevent the dreaded 5MWD.
And, again, the Cleric in 5e has far too many spells, other class abilities, and even melee to fall back upon, to be 'only' useful as an in-combat healer - and the Cleric in 4e had plenty to do buffing and contributing directly, that (minor action) healing was hardly his 'only' function. You'd have to go back to AD&D to find a 'healbot' cleric.
I know that, personally, I'm going to have to tweak the natural healing rate to allow for attrition over the course of several weeks rather than giving everyone a free reset every night. I'm not sure what rules will be in place for that, or if it will be as simple as dialing up the length of short and long rests. The stronger the rely upon their base assumption, the less room there is to accommodate such tweaks, and those are supposed to be the most important part of this edition.
It's a save assumption that the DMG will have some sort of guidance. If it's about /pacing/ - if, for instance, it advises you to change the definition of short or long rest (short rests are 8 uninterrupted hours, maybe - long ones days of R&R in a safe/friendly environment, perhaps), then you follow that advice and get a slower-paced campaign where a day's worth of encounters can be spread over a week and things work out reasonably well. If it advises you to do that /just/ by changing the rate of natural healing (X hps/1 HD/whatever per long rest or something), then following that advice could seriously impact class- and encounter- balance, as rest-recharge abilities become much more available.