Estimated cost savings by going from a 2 layer to a single layer pcb?
Single layer boards are not usually made in the same facility as double and multilayer boards. The materials (most often cheap phenolic boards) and techniques (punching as opposed to drilling- so many huge punch presses for holes and outlines rather than drilling and routing machines) are quite different and the cost per square meter (yes, that's how single layer boards are priced) will likely be dramatically lower than anything you can find online, particularly if you design to what is most used rather than trying to get them to do something special. Shipping by surface (sea) is most economical. 

The design has to be more careful to use the cheaper fragile paper-based phenolic laminate- much larger pads, no heavy parts mounted to the board without extra support and so on. They are not nearly as tolerant of abuse as epoxy-glass boards with plated-through holes. Parts the user can interact with such as connectors and encoders should have support provided elsewhere or the product may not be reliable.   

Even with hard tooling, the minimum quantities may not be that insane- maybe 10K pieces (per shipment) for a board of reasonable size. Of course there will be NRE cost for the punches, which would typically be applied to the next shipment (provided the maker is still around and you have not made any changes to the PCB). 

If you are using FR4 and quantities are smallish there is little advantage in single-sided- you can probably make the PCB smaller with double sided and recover whatever meager discount the PCB maker would give you.