In this answer, it claims that the reason for lift decreasing at the wingtips is due to airflow around the wingtips. If I'm thinking correctly, that is due to the air below the wing spilling on to the top, decreasing lift (correct me if I'm wrong). That made me wonder if adding winglets evens out the pressure distribution along the wing, because less air spilling over (from bottom of wingtip to top) would increase lift, right? In that case, say you went to an extreme, and made a massive winglet, would that basically get rid of tip vortices and make the pressure distribution almost completely even?
Winglets do change the lift distribution on a wing.
To really take your line of thought to an extreme, replace the winglets with flat walls that extend up, down, forward and aft to infinity.
If you place a rectangular wing in a wind tunnel such that the span is equal to the width of the wind tunnel, you can simulate two-dimensional flow with a uniform lift distribution across the wing.
Most high quality airfoil data has been taken in a tunnel set up to use this effect.
Of course, winglets also add weight and wetted area (skin friction drag). So you don't want to install a giant winglet on an aircraft.
Whitcombe originally called winglets "tip sails", because they operate much like sails on a boat. They exploit the energy in the tip circulation by placing an airfoil surface such that lift is generated inboard. Since the circulation flow is moving inboard, it has a positive angle of attack to the winglet and the lift force is perpendicular to that angle of attack. The lift force is pushing the winglet toward the fuselage, but also angled forward somewhat. The forward component of the lift force is, literally, thrust as far as the airplane is concerned. Winglets make thrust, like a sail boat close hauled.
While making that thrust they are consuming the energy in the circulation and creating an outwash (downwash sideways) that opposes the circulation, weakening the core of the vortice and having an effect similar to a span increase.
The end result is the lift distribution is improved similar to a span increase, and some thrust benefit is also extracted from the flow that may exceed the winglet's drag.
They only generate enough energy to compensate for their own drag when the wing is "working hard" operating at higher AOAs, that is, low indicated airspeeds. The only airplanes that cruise at low indicated airspeeds are jets that fly above 30000 ft (for example, 443kt TAS at 37000ft is only 250 kt indicated). This is generally why you only see winglets on jets, and putting them on straight wing low altitude aircraft is mostly for looks, or to improve rate of climb (where AOA is relatively high enough to get useful work out of them).