I don't play 5e as an attrition game, at all, so I have design the encounters that do happen very much with balance in mind, taking into account that the party will likely be starting with almost full resources. Encounter design has a massive impact on balance, and when I screw it up, I really screw it up!
Sure, that will certainly happen if the PCs rest enough that they approach most encounters at full strength (or close to it)!
But that that point, like the other poster's examples, just plan on every encounter being at a minimum Hard, and often Deadly or worse.
For myself, I've found my players nearly have heart attacks when the encounters are deadly (as an attrition game!). Their resources get depleted, but there is still more to do. Can they rest? Do they dare leave and try to come back? Who knows what else will be waiting then!? Can they handle the "next room" or not?
In our game Friday, my DM PC died. The party was four 6th-level characters (Rogue, Fighter, Paladin, Sorcerer/Cleric) against two sea hags and four sahuagins. The adjusted XP total makes this a Hard encounter, which it was, and dumb luck (failed a save by 1) droppe my PC to 0 hit and in the clutches of a sea hag. A failed death save followed by a critical hit on a downed PC for a double-tap ened his adventuring career.
The next encounter was now 3 against 6 (sahuagin coral smashers). This was now (with one less PC) a "beyond" Deadly encounter. It nearly resulted in a TPK at that point. The party was decimated in the end and fortunate that it was the final combat of the adventure.
All in all, I have found the attrition method works well
if your game can play that way. Otherwise, yes, you certainly will have to ramp up the difficulty to a minimum of Hard IMO.
My only points are that A) what's deadly for one group is only moderately dangerous for another group and B) it's always going to be up to the DM what difficulty a group can handle.
Fine, I wished you had expressed that a bit more clearly. However, in my experience I have found A) is not often true in an attrition-run game (things tend to even out) and B) a lot of DMs misjudge what a group could handle.
There is no one size fits all and the encounter rules in the 2014 DMG are aimed at the low end. For some groups I never throw anything other than hard fights and we generally have 4-6 fights between long rests (occasionally more, occasionally less). Other groups? I'd be worried about a TPK by the second fight.
I think they are aimed at the attrition end, not a "low end".
However, I know since I do encounters based on the story and not a game-enforced design that often hard encounters won't be hard, but after sufficient attrition a medium encounter might become deadly or even a TPK. I've had times when I had well over a dozen encounters between long rests, and many more times when I had only one. For me, the story drives that, and I don't expect by any stretch of my imagination that most encounters will be (or should be at a certain point!) challenging to the PCs. It is the series of encounters over time which results in the real tension rising!