Here we go again, this is the 2nd gravity wheel I have posted on in a day, actually the one you describe is very similar to one I analysed about 20 years ago, that particular idea used an eccentric to move the weight closer to the shaft on the up motion. First off gravity doesn't care less about motion orthogonal (horizontal) to it's axis of action so the side to side motion of the weight can be neglected in this analysis, the weight always starts at the same place as it finishes which in a system with constant force regardless of position means no net work is done. For example take a 2kg weight with a 1m fall top to bottom, going up the weight rises 1m needing a work of 2kg x 9.81N/ms x -1m = -19.62 J to raise the weight the weight falling back to the bottom which is also 1m (it is impossible for it to be anything else) gives work of 2kg x 9.81N/ms x 1m = 19.62 J exactly the same as that required to raise the weight.

Just to be clear on why the distance the weight rises is exactly the same as the distance the weight falls if it were otherwise the weight would start rising or falling with each rotation of the wheel and leave the wheel perimter. Still not convinced, draw a 1ft circle and mark the top and bottom Draw a line though those points representing the vertical motion on a real wheel. Lets say the weight rises 11" but falls 12" in this situation assuming it were mechanically possible the wheel will spin. Now measure 11" from the bottom of the wheel along the vertical line you drew and mark that point then measure 12" down from that point and mark that point, it will be 1" outside the wheel. imagine that for every rotation of the wheel the weight would end up 1" further below the wheel. This can be made to work in practice, wind a cable around the outside of a wheel with the weight attached, as long as there is cable left and space for the weight to fall the wheel will spin and be capable for producing work the only problem is that there is no way of getting the weight back to the top without putting in a similar amount of work

With gravity it is only motion in the vertical plane which can produce work

There is another way. What if your weight was not solid. Lets say it is a liquid in a small tube that can slosh back and forth. You could try radiating out from the center of the wheel or other angles too. Yes, this has been tied before too, but how now about if it is not water but say a ferrofliud instead. You could use fixed magnets on your stator that will change the position of the fluid as the wheel spins. you can get the input power from the falling fluid on one side, mitigate the negative energy of the rising fluid on the other and do it all without any loss due to friction. The only problem will be using the fluid and magnets in such a way that you can minimize the magnetic attraction\repulsion on the fluid. Search around for water wheel contraptions.

I know a gravity wheel could be made that will produce circular motion, but I don't know if enough meaningful work could be produced.  It will spin (spin sounds like it means a fast rotation and speed is not truly what I expect) but almost has to be in a frictionless environment, thus may not work under a load.

We start with two spokes one at 90° and another at 45°.  We have a heavier weight on the 45° and the lighter weight on 90° spoke is 66% of the heavier weight.  The weights are connected by a pulley system so as the heavy weight moves towards the axle at 45° the lighter weight moves away from the from the axle at 90°.  That's on the top, on the bottom the angles are technically the same (really 270° and 225°) and the weights move in the opposite direction.

The distance moved by the weights is the same.  The heavy weight is fully extended 75% on one side and 25% on the other, while the lighter weight is fully extended 100% on one side and 0% on the other.  What's more important is that on one side 25% of the time both weights are at the closest to the axle and on the other side 25% of the time both weights are furthest from the axle.

A latch, just like a door knob, is needed to ensure the heavy weight doesn't start to shift until it reaches the 45° angle where the rotation then turns the latch. Additionally, once the heavy weight has been released and while the weights are shifting locations the rotation of the wheel should be paused. 

To ensure there are always spokes inside the unbalanced area you want to have more than eight separate pulley systems in the wheel.  The amount of imbalance between the two sides of the wheels is not that significant, so the mounting for the pulley systems needs to be as light as possible, as not to negate the amount of imbalance.   

Different size weights (percentage of difference) could be used, which then requires different angles, and then a different number of spokes.  The larger the distance the weights move the better, and in general more spokes are better.   Also just changing the angle (or the difference in weight) affects wether the pulley system or the rotation can handle more load.  Since the pulley system should have less friction the angle could be changed to 42°, increasing the rotation ability. But at 41° the pulley system won't work even if it was frictionless.

It only works when you add a force that is independent from "the world". This means, it may not be fixed/based on the stator, but on the rotor.
The Water Motor (see left frame) is an example for something like that. Moisture is deforming some stripes on the rotors outline, causing them to bend towards
the center. This makes it a gravity imballance wheel. The bending stripes haven't got any physical or magnetical contact to
the stator, that's why it works. They don't have to push themselfs away from something, but simply deform by their own "power" (whereever that comes from).
But this is just an example.

Simple magnets may be the wrong choice, since the act like any physical contact from stator to rotor - you could even push the wheel with your finger instead.

Turbines are wind asisted or water asisted gravity wheels using the forces of nature including gravity to produce power ... I think the key is to make exsisting devices more easily produceable for manufacturing and useable for the end user aka saving time and money all around.