If you want to sail downwind you don't need a keel except for balance. In land sailing and ice yachting the wheels or runners serve the same purpose as a sailboats keel and provide resistance to the vessel moving sideways in a crosswind.
A boat's keel is weighted with ballast (on larger boats) for the balance reason, but it's fin shape is for the purpose of providing lateral surface area.
Even while sailing downwind, the sails are still providing "lift" (if we are calling it that here). There are opposing forces for sailing - the Center of Effort (CE) and Center of Lateral Resistance (CLR). These two need to be balanced in order for the boat to sail a proper course - of course if you are just letting the wind push you directly downwind, you could remove your keel/daggerboard, but in practice, you will want to sail on an angle to the wind - on a reach or by the lee depending on the type of boat - so as to create flow over the sails and have a greater pressure gradient than if you just let the wind push directly against the sail.
The hull of the sailboat contributes to the CLR, so it is common in dinghy racing to pull your centerboard/daggerboard up partially so as to balance these forces and minimize drag. If you watch Olympic laser sailing, you'll see while sailing downwind, they will heel the boats to windward with their daggerboards about halfway up so as to move the CE closer to being above the CLR, allowing for less rudder drag to counteract the forces and steer straight.
Every time there's anything on HN or Reddit about sailing, there's a _ton_ of misinformation and blanket statements that might apply to some types of boats, but are far from general rules. I'm happy to answer any other questions too!
I was trying not to overcomplicate the explanation, but when I said "downwind" I meant dead downwind, not broad reach. In this situation your sail plan does not need to generate lift: You can fly a symmetrical spinnaker and be propelled entirely through its drag forces, at the cost of not being able to match or exceed wind speed the way you can with a mainsail generating lift.
A boat's keel is weighted with ballast (on larger boats) for the balance reason, but it's fin shape is for the purpose of providing lateral surface area.