Brakes that vibrate when engaged, worse at higher speeds indicates the rotors are warped. This can happen for many reasons, hot brakes and drive through a cold water puddle for example. If the steering wheel shakes when you brake, it is the front brake rotors with the issue.  If the steering wheel does not shake it is probably the rear rotors. The mechanic that looked through the wheels and said you didn't need the rotors machined was probably saying there was no unusual wear and/or the brake pads have not scarred the rotors. The only way to determine which rotor (or rotors) is to blame is to use a dial indicator to measure the rotor face while turning the wheel.