The trade off would be that your A-D converter would have to be fast enough as well as offer enough bits of result to give the iterative algorithm a running start. Digital video cameras would be the class of device we ~all have that does high-samples-per-second conversion. A 4K@60p camera has to do 500Ms/s at least so if you can allow that many approximate results per second then it might work out. There would also have to be a comparably fast DAC to tell the analog circuit what you want to approximate.

I think a more-complete computing engine in analog would be a bigger win, because there won't be conversion overheads. Or as was suggested somewhere in this topic, a field-programmable analog array.