Thanks go to RockPaperShotgun for pointing me towards Seedling, a delightful little Flash game on Newgrounds that evokes classic Legend of Zelda gameplay (A Link to the Past in particular) in a pretty great way. Controls are extremely minimal, as Flash game controls are wont to be, but Connor Ullman manages to cram a lot of good exploration, item collection and puzzle solving into what he's got to work with.
From his website:
Seedling (renamed from “Shrum”) is an adventure game in which you play as a small boy who is tasked with finding a seed to replace an enormous tree for your creator, the Oracle. You will fight many monsters, collect many tools, and eventually find a way to get to the seed—at whatever cost that may be to yourself and this old land you have changed so much along the way.
...Seedling features over 100 areas spread across an overworld with eight dungeons, six different weapons, and several other types of items that give the player new abilities that will help them traverse a world that varies from icy wasteland, to heavy forest, underground caverns, floating clouds, and magic castles. Boss battles and enemy grunts will meet the player at every turn as they try to gather the tools necessary to reach the seed; a goal that will destroy beings known as the Creatures of the Relic, all while under the eye of the Watcher, who sees and judges your every action.
Here's a trailer, which should make the aesthetic inspiration very clear to anyone who owned a Super Nintendo. (EDIT: @lavos quite correctly points out that Link's Awakening on the Gameboy might be a better reference point for the visual style. I think I have Link to the Past on the brain due to a recent jaunt through the Wii Virtual Console.)
I've only put about a half hour into it so far, enough to get the starting items and explore a bit into the landscape, but there's a whole lot to like here for a free game you can play in your web browser.
One notable change from the classic formula - your health regenerates every room, so if you die, your loss of progress is extremely minimal. I'm not sure that sort of mechanic would serve Link to the Past, but Seedling is twitchy enough that it's nice for it not to punish you too harshly for the occasional failure.
I'll be going back to it for sure. If it sounds like your thing, go give it a try.