Just curious: How many here have played A Link Between Worlds? Would that game be a step in the right direction? (Giving you almost all items from the get go, freedom to explore)
I think ALBW is a good first step mixing the freedom of accessing a great part of the overworld since the beginning of the game, interesting things to do almost in every screen, and the ability of doing the dungeons in the order you want.
But it has some flaws, too: the map is 99% identical to ALTTP so when I started the game I already knew almost all of it (the sense of exploration and discovery secrets is damaged), and the dungeons are too easy (they should tier them in three or four difficulty levels so you can do 1-2-3 (easy ones) in any order, 4-5-6 (normal difficulty) in any order, 7-8-9 (hard ones) in any order and lastly the final dungeon that is only accessible when the others are completed. Other option I like would be let the dungeons with its own difficulty accesible from the start (like in the first LoZ) and leave it to the player, so he can explore and try it or leave it if it's too difficult for his actual level/itmes). That would need them to implement what LegendOfLex explained about dungeons being less puzzle-like and more maze-like, and less dependent of the items you get.
I prefer to get the items not from the start like in ALBW, but also not only in the main dungeons, which is becoming way too predictable. Maybe with NPCs and side stories, mini-dungeons/caves (the not essential items, but powerful ones). I think it's fine if the game starts with Sword, Shield, Bow and Arrows, and Bombs. And those being upgradable. So you can start doing basic combat/exploration.