Laying bricks over old concrete sidewalks
If the old slab is still “sinking”, then it will continue sinking after the new bricks are installed on top of it. In fact, the additional weight from the bricks may accelerate the movement. 

I think you have 4 options: 1) keep the slabs and do not add brick pavers, 2) remove existing concrete slab and install new slab or brick pavers, 3) mortar new bricks to existing slab, or 4) dry-set bricks over existing slab. 

1) If you keep the slabs, you’ll need to find the cause of the settlement...sooner or later. It doesn’t sound like it’s a big deal now, but could become unsightly soon. Finding the cause could be difficult, unless it’s obvious, like drainage swale eroding, etc. 

2) Wow, removing all that concrete is a lot of work and it sounds like most of it is just fine. (One plus for this option is that you’ll look like Charles Atlas when you’re done.) This is the most expensive option too, because you’ll need to replace it with something. 

3) Mortaring bricks to the existing slab is least desirable, because you’ll need to fix the low spots, adjust the layout and install all the joints in the bricks so they align with the joints and cracks in the slab. Otherwise, the new bricks will crack randomly. 

4) This seems the most reasonable. Installing dry-set brick pavers on the existing slab, you’ll first install a layer of sand for the base. This base can be adjusted for low spots. Also, in the future, if the slab settles more, the pavers can be removed and additional sand can be installed and then the pavers reinstalled. HOWEVER, the sand base must be contained. That is to say, the sand under the pavers and between the pavers will wash away if it’s not contained.