Increase drain size?
The sink drain you show in the picture is a standard kitchen drain that meets code. Its only purpose is to drain water. Based on your post it seems that you are getting clogs in your drain at the mouth of the drain as shown in your picture. It may not drain fast enough for you but it is doing what it's designed to do, trap large debris before it passes down the drain resulting in a potentially costly repair. To make the openings any larger could result in a blockage issue.
It's also a good idea to use the strainer that fits in the drain. It's much easier to clean out the strainer than to have to call a plumber.