The buttons are larger than half an inch. Cuff buttons are more than half an inch (traditionally 23L-24L or a bit less than 5/8"), and the buttons on the front of Grant's jacket are larger than typical cuff buttons, so they are much larger than 1/2". His front buttons look to be roughly 28L-29L (23/32"), which is on the small size for a single-breasted jacket. 29L or 30L (3/4") are traditional single-breasted sized buttons, while 32L is the most common suit jacket button size today because it can be used for both single-breasted and double-breasted. That is why these buttons look small to you. Double-breasted jacket buttons are traditionally from 34L-36L.
For a single-breasted jacket, I'm happy with anything from 29L to 32L. For a sportier suit, I like it on the larger end.