the design is intended to use the vortex to generate thrust in the forward direction
I would be a little wary of using the term "thrust" when describing the effect of winglets. I know I may be debating semantics, and that with a thrust-drag force couple, reducing one is roughly equivalent to increasing the other, but I would still think of the effect of winglets as
reducing induced drag, not
increasing thrust.
Can we turn off the engines and fly using the "thrust" produced by the winglets? Would those winglets be producing any force if the engines weren't propelling the aircraft through the air?
In aerodynamics you can't just look at a single component and say it produces thrust. You have to look at the interaction with the neighboring components. If you look at a wing with a winglet installed it's obvious it does not produce any more thrust than a wing without winglets, or a landing gear or an antenna sticking out of the fuselage for that matter.
I'd say to leave the thrust part to the engines
The short and sweet answer to the original question is that the net effect of winglets can be approximated by hypothetically increasing the wingspan in the amount equal to the height of the winglets. What the winglets accomplish is basically "fake increased aspect ratio".
If instead of building the winglet you were to increase the wingspan by the same length, you would achieve approximately the same reduction in induced drag.
And even this effect requires a very carefully engineered winglet... otherwise the net effect could be a performance loss (or, more often, a smaller reduction in induced drag than what could have been achieved by an equivalent increase in wingspan).
Depending on the design, a manufacturer can opt for one option or the other. As you can imagine, there are advantages/disadvantages (ie. costs) associated with either option. With the increase in air travel and subsequent saturation of most airport hubs, the reduction in wingspan allowed by winglets can be very useful in fitting more aircraft side-to-side in a given terminal. For example a CRJ-200 has a wingspan on 21 meters, with each winglet measuring approx. 1 meter. That's almost a 10% reduction in wingspan thanks to winglets (ie. every 10 aircraft you can fit an extra one at the terminal.) Do not underestimate the importance of this aspect.
What always amazes me is how the majority of aviation enthusiasts (including many professional pilots) think of these winglets as the latest-and-greatest invention and that just slapping an end-plate (or winglet) will dramatically reduce induce drag (some ATPL books even show a winglet "stopping" any vortex generating at the wingtip). Its not this simple nor this effective. If that were the case all aircraft would have them
Lastly, let's not forget that marketing plays a role in every product's R&D, including aircraft. It's no secret manufacturers have been trying to find every possible gimmick to differentiate their product...