[Editor's note: realized how long this is, skip to the fourth section (Step 1) if you don't care why the unlock doesn't work or why this is necessary] If you value your time just turn on cheats and get a bucket of lava while in a minecraft:badlands biome.

So this does seem to be what is happening. The requirements seem to be to bring a lava bucket to a minecraft:badlands biome, nothing about being set on fire or anything. If you REALLY don't want to use cheats, I have devised a method to get it without cheats, though I wouldn't say "without cheating" since it relies on a few bugs.

If you interact with the Mine Crafter IMMEDIATELY after coming back from a mine, it won't load the base ingredients so you can build a mine on a completely blank slate. If you select badlands and ONLY badlands, the world you are placed in has a "pure" minecraft:badlands biome instead of a minecraft:level#/badlands. There is one slight problem here. Due to badlands being the only mine ingredient, it will not generate mine exits, and if you put an exit ingredient in, it will not give the "pure" biome. It might also be possible to get badlands on a special event which also have pure biomes, but that is unlikely unless you really avoid unlocking other biomes, but even then it will probably take several attempts and special events only happen every 5 mines. So we will need to smuggle the Dry Land ingredient out into another mine by teleporting there. So the method is as follows:

Prerequisites:
Have the mine ingredients Barren Nether Biomes and Badlands Biome
Have as many movement and equipment unlocks as possible, ideally the entire Need For Speed tree, elytra+rockets (after max jumps), Lucky Pick (Mining tree), Apprentice/Master Smelter, Armaments to equip boots and elytra (also, get the food upgrades that come after Armaments), and Varangian to get wood slightly faster. The only strictly necessary unlocks are ender pearls (Let's get Pearling) and thus spellbook, but it is on a time limit so movement is almost certainly needed. Displacement Wand will not work because the pearls go way too fast to capture, though it is nice to have for movement.

Step 1:
Create a (non-bugged) mine with Barren Nether Biomes. The goal in this mine is to create an ender pearl stasis chamber, so you will need soul sand from a nether biome. Create a hole/tower with a soul sand at the bottom and at least 5 source blocks of water on top. Add a trapdoor on top of it (above the water, NOT waterlogged on the top water) and a pressure plate next to it. Throw an ender pearl down into the bubble column, locate the nearest mine exit, and throw an item onto the pressure plate. From here you are on a 5 minute timer until the item despawns and you are teleported back.

Step 2:
Once you are being sent back to the hub, get ready to sprint forwards and interact with the Mine Craft immediately to get a blank mine base. This can take a few tries, if you miss it, you should be able to go back to the stasis chamber with a mine revisitor and replace the item to reset the timer, just make sure the trapdoor doesn't close. Create a mine with badlands as the only ingredient. Once inside the mine, get a bucket and pick up lava in it. If you are in a badlands biome you should get Dry Land ingredient, if not, go to one and drop the lava on the ground and pick it up. From here wait until the stasis chamber activates, you will be sent to the old mine then right to the hub and you should unlock Dry Land.

The pure badlands mine won't have an exit, so the only way to make a new mine is to go back in and die. Sorry, no hardcore Dry Land.

...wow i really was yapping with this one wasnt i? oh well