You didn't mention what the distance was from the edge of the pool coping to the fence, so I don't know if it is a large area or just a short distance. Regarding your husband's concern of the river rock not staying in place, that would depend on how steep the slope is. If it is a reasonable slope of about 4:1 or less, I think it should be fine. A 4:1 slope means that, for example, the following. If you have a distance of 4 feet, the difference in elevation would be 1 foot from edge to edge.
Would you consider as an alternative a low maintenance ground cover? Ajuga is a bit aggressive, takes the sun or shade, and is attractive. If you planted it on the sloped areas and added a thick covering of mulch, you should not have much of a weed problem. You can also put down some weed preventatives. Back to the stone, I do like river rock.
