Or, to put it another way, I'm not looking to make attrition workable. I'm looking for a way to make fights dangerous and fun
without the use of attrition.
Youll have two main problems doing this.
1) It screws with class balance. Big time. Longer fights and shorter AD's favor long rest dependent classes (full casters, paladins and barbarians). They screw over short rest dependent classes (warlocks, fighters and monks).
Assume a warlock 9 and a wizard 9 are in the same party; one that features one 'big' deadly battle per AD. The 'lock gets 2 x 5th level slots to ration over that one battle (and there is a good chance the 'lock is gonna use one on Hex so his cantrips dont suck - he'll be needing them). The wizard meanwhile gets 1 x 5th level slot, 3 x 4ths, 3 x 3rds, 3 x 2nds and 4 x 1sts to nova with. Clear advantage to the Wizard.
Now assume a 2 short rest/ 6 encounter adventuring day.
The warlock now has 6 x 5th level spell slots as his AD resources, competing with the wizards available spell slots - yeah the wizard still gets more slots (15 compared to 6), but the 'locks spit out at 5th level, and he has his nifty pact features, and at will invocations and a better than average cantrip to fall back on. The wizard only gets the 2 slots (factoring in arcane recovery) of 5th level, 3 of 4th level, 3 of 3rd level (and some 1st and 2nd level spells, that are about on par with the 'locks invocations). 30 spell levels [plus invocations] for the 'lock to distribute among 6 encounters, vs 41 spell levels for the wizard. The classes balance out.
Now lets look at a champion fighter v a vengance paladin. They both get the same number of attacks. The champion has a better crit range of 19-20, the one action surge, the one second wind healing around 15 damage, and the one indomitable. The paladin has lay on hands (heal 45) channel divinity sacred oath option (advantage for 1 minute) + cha to all saves, auras, spell buffs, and can smite nova up to ten times dealing anywhere from +2d8-5d8 damage per attack. Clear advantage to the Paladin.
Again, now look at those classes over a longer (6 encounter/ 2 short rest) AD. The longer the AD the more the Champions 19-20 crit range comes into play more often - more attacks = more uses of the ablity. He now gets 3 action surges (each one allowing generally 2 attacks, probably dealing around 13 damage each - about on par with 2 x 3d8 damage smites). His second wind is now useable 3 times, adding up to 45 damage worth of healing - again on par with the Paladins lay on hands. He doesnt have the spell casting of the Paladin, but now the Paladin has to ration out those 10 spell slots over six encounters, and not just the one - while the Fighter can rely on the 'always on' nature of his bonus feat and remarkable athlete to more or less cancel that advantage out. The classes more or less balance at this point.
When you look at the math of the system, it tends to balance around a sixish encounter [each of less than 10 rounds]/ two short rest adventuring day. The classes all more or less balance at this assumption. Altering the structure of the AD affects this class balance. I know you dont want to alter the mechanics of the system (either by changing the rest mechanic, or by buffing many of the classes) but if you dont, this problem is unavoidable, and the classes will not balance, and many will rapidly begin to outshine others.
2) You are dramatically increasing the risk of a TPK. Youre encouraging the party to use nova tactics, and resort to the 5 minute AD. Like I said above, this drastically favors full casters, just like it did in 3.P. They'll nova like madmen on round 1 of your encounters, and to compensate youll have to dial up the difficulties even further by using higher CR enemies (many of whom have abilities with DCs or damage that could easily kill a PC with a single flubbed roll). Rocket tag ensues, and sadly as there are always more monsters (and no-one cares if they die0 increasing the lethality of your game, punishes the players.
I strongly suggest using the longer rest variant if you want to do this. It cuts down on nova strikes, and ameliorates many of the problems above. PCs wont be novaing long rest abilities (those long rest recharge ones are always the good ones) when they know they need a week to get them back. In return you only have to dial up your encounters half way (and not go full abyssal horde on the PCs), meaning your encounters wont grind, balanace will be maintained, and tactical choice (risk reward for hitting the long rest resource button) becomes much more of a meaningfull choice.
You lose nothing by using the longer rest variant, and its a win/ win.
If healing bothers you, healing potions are a thing. PCs dont have that much to spend GP on in 5E anyways. Make them accessable to the party.