I'm building a bike from scratch and old parts for home-trainer and roller use only. Do I need brakes or can I cheap out?
No. For fixed trainer, it should be obvious.
For rollers, rolling resistance is higher on rollers than on flat surface. Wheels don't also have much momentum, so you can slow down by freewheeling for a few seconds. The only use for brakes is preparing to dropping off the rollers: rotating tires leave a nasty skid mark on the floor but brakes stop the wheel immediately.
I find brakes help with stability on rollers. When mounting/dismounting hold the brakes so your bike does not roll off the front or back of the rollers. Additionally, if you lose focus while spinning and begin to lose control, you can more easily correct when applying the brake.
As also mentioned, if you roll off the sides, brakes will help to immediately stop the wheels and avoid toppling.