These legs and aprons serve to support and will also make the island look much better (not look like a diving board for lack of a better word) - and the cost should be the same as if it had been added to the island in the design stage. If it is more expensive than that (may be due to the separate manufacturing of the legs and aprons), then the kitchen designer should eat the difference since he/she should have incorporated the support for the slab in the plan. The contractor follows the plan and has no responsibilities other than proper installation.
I have an island similar to the one in the picture with cabinets under like that - we only store once a year stuff in them as accessibility is cumbersome at best. With these legs in place you should be able to open the doors (22"?) at the ends wide enough to get stuff in and out.
Good luck and be good!
Q
