The thing you're suggesting that departs from the Classic model (but also brings it closer to the Conquistador one) is the backing of a wealthy patron or patrons. I think there are pros and cons to this, but I think it could be great fun.
For this sort of campaign, I'd find a good set of "wilderness events" or "expedition events" random tables that you can roll on. E.g., disease starts going around, or wild animals eat some of the rations at night, or you get slowed by weather or terrain, etc.
Then I might try to abstract out things like "food" and "equipment" and "treasure" into levels that can go up and down based on the character's decisions. Like if you go out hunting your food increases, but if you explore the dungeon your treasure increases. Maybe you find an old supply cache and equipment increases. Then, random events force you to select between these resources. Like if you encounter a river you need to cross, you can build rafts (which uses up some equipment) or go around a longer way (which uses up some food). The goal of such abstractions would be to make the PC's expedition-management decisions more interesting by clarifying their impact, while keeping things abstract enough to play very quickly.
I like the idea overall. It has a feel to it like in some movies like Indiana Jones or even older movies where they search for lost cities of gold. An expedition to a far away land removes the players from the known and places them in another land, like Egypt for example. Getting there can take several weeks of play with setting up ships and terrors at sea. Npc's that sneak aboard ships and ones with hidden agendas or ties to another evil group wanting the treasure, or prevent someone from getting it (the movie The Mummy). Treks across the desert or jungle have natural hazards and monsters unique to that environment. Local savages may help the characters in their quest.
The extra npc's that come along all can have more role to play than mundane. A dead character can be replaced from this pool rather easy. A bad guy can emerge who tries to con locals into hurting the characters. These npcs also have skills like smithing and arrow making that the party may need. One can be a sage or sorts who knows a lot about the region. A guide that once visited here. 10 years in Dragon Magazine there was a campaign based on the Isle of Dread which has a lot of similarities.
Don't forget to end the major encounter with lava explosions and falling temples with limited time to escape.