A rimmed sink can be secured in a bed of silicone IF it's done correctly. Judging by your pictures, it wasn't.
Here is what I would do. First, use a razor scraper to completely clean all the existing caulk debris off of the top of the counter. Scrape and clean the bottom of the sink lip, also. If this is a stainless steel sink you can use sandpaper, and Scotchbrite pads to really clean the underside until it shines. Next, use acetone and a rag to scrub both surfaces to remove any residual crud. Now dry fit the sink in the hole. Use masking tape to cover about an inch of the countertop, placing it tight to edge of the sink lip. Flip the sink over on a flat surface and put a very large bead of clear silicone on the bottom of the lip. Lower the sink into the opening, and evenly push it tight to the counter. Clean all the excess squeeze out from the edges and place a good amount of weight in the sink ( a few small buckets of water, stack of hardcover books, etc....) peal the tape up, check for any caulk mess, and let it sit overnight. If everything was done well, this sink will be there until you pry it out.