A Playstyle Question, Not A Balance Question
While it is entirely possible and relatively simple to design a balanced form of "day-based" resources, it is not something that every D&D player appreciates. Some D&D players play in a more scene-by-scene style, where they pay attention only to what abilities the party can call into a single scene, without worrying about different recharge rates or resource management.
It's important to note that this isn't necessarily a better or worse way to play. It's a different way to play. Traditional D&D has elements of long term resource management and accounting as a way to embody a slow whittling away of resources, a gradual degeneration of security. There are those that wish to minimize this aspect, out of concern for their own preferred pacing.
There's definite advantages in this style. It makes each encounter more important, it weights immediate action, and it can lead to a more constant high-tension style, where each encounter is decidedly key. It is, however, markedly different from traditional D&D assumptions of daily journeys and week-long dungeon delves. This requires some alterations.
Encounters Within Adventures
It is possible to have an encounter-based game within a game based more on the adventuring day or adventure cycle fairly easily, simply by adjusting a few expectations.
First, Everything Comes Back After Each Encounter
The end of each encounter is treated like an extended rest. This is similar to Time Shifting Rests, but rather than making the extended rest longer, you make it shorter.
As an analogue to this, your "second wind" is actually a short rest. You can take a second wind at-will.
Second, Each Encounter Is An Adventure
Each encounter you do has to have its challenge determined as if it was the entire adventure. If you use a 4e model, that would mean that instead of having 1 monster per character over 3 encounters for the day, you'd have 3 monsters per character in one encounter. Or the equivalent.
Keep In Mind For Design
If you're going to enable your game to be played in both Daily and Encounter-based mode, you're going to want to inhibit the ability of the party to negate encounters, and you're going to want to keep every party member able to contribute to every encounter. You can either bake these into the core rules, or you can allow them as part of an "encounter-based game" package. Mostly, I'm in favor of the former, since that makes the encounter-based game easier to apply to the base game, and you can always take away these elements with other modifications: a bard who is useless in combat might be fine in a daily game, but it wouldn't be in an encounter game, so if it's something you can opt into, it's something that you don't have to be stuck with (and you don't have to change) in an encounter game. And you don't HAVE to change it for a longer game, you just have the possibility of it.
Advice: Diversify Your Encounters
Any game that is based on the encounter is going to spend more of its time in each encounter than a game that is more broadly based. Thus, it's important to keep in mind how to diversify them, to add different kinds of threats and different kinds of obstacles. Elements like in-combat traps and terrain elements and minions, standards, elites, and solos become more important when you have an encounter-based game than not.