Any dedicated track frame should match your desire. They tend to be pricey. Does he want steel or aluminum? EAI Bare Knuckle is a top quality steel frame with no holes, other than in the fork crown and seat stay bridge, for brakes. They are no longer in production and possibly only available at Retrogression--Google it. Another option is the Cinelli Vigorelli Steel. (Did I mention these choices are expensive?) Any used Keirin racing frames will be hole-free and high quality, but maybe not in great shape cosmetically. If you go with aluminum, prices are a bit lower. Dolan has some good offerings, and so does Aventon.
The braking question depends a lot on what kind of riding he's doing. On flat terrain at moderate speeds, a coaster brake is sufficient. Riding at higher speeds, in heavy traffic, or down big hills, I think a good brake of some sort on the front wheel is almost mandatory.