Overlay would be done with Acrylic which if a good material is being used can be much stronger than concrete. You can get Acrylic that is 8500psi at 1/8th inch thick. Concrete is only 3600psi. So in the long haul Overlay should last a lot longer. Also gives you the opportunity to change if you would want to. If you stam,p the concrete and it cracks as they all do eventually, you are pretty much stuck with it. Overlay can be fixed, so I would have to agree that overlay is better.

On the front elevation I'd get rid of the gable over the garage. All it does is lead your eye to the garage and detract from the main body of the house. It's not a feature. The other thing I'd do is insist on them not doing the pork chop returns. Your house has a historical feel to it and that just screams contemporary mass builder.Below the red arrows point to the pork chop returns. Lastly on the rear, you have one window that isn't consistent with the rest of the windows. The one with the arched top. I'd change it to a transom window top to match the other transom windows. BTW: What is above the window I changed? That's a large blank wall on the exterior.

That your existing concrete is in good shape is key. Any crack in it would transfer through to the new material. So long as you create proper finish levels, you should be in good shape. Stamped concrete can look terrific. It's biggest drawback is the color not being integral to the slab, but just surface treatment. This is really no problem unless it is subject to harsh wear, which in this instance is probably not going to be the case. If the surface gets a chip or gouge, you'll see near-white concrete below. In most residential situations, it doesn't get that harsh of treatment.

You would be able to do it if you level the concrete out and then do the pavers. However please keep in mind this would increase the Hight of your overall backyard. I would still recommend demo of the concrete before adding pavers. Hope you have a wonderful rest of your day!