The extension would only need one initializer:
public init(randomByteCount: Int, using generator: RandomNumberGenerator? = nil)
But either way, how would such an initializer be implemented? Should a next(_ bytes: UnsafeMutableRawBufferPointer) method be added to the protocol? This could be in addition to, or instead of, the generic next<T: FixedWidthInteger & UnsignedInteger>() -> T method. It could even replace the next() -> UInt64 method, but in that case other generators might need to use an internal buffer.
Another suggestion is to rename Random to DefaultRandomNumberGenerator. Its property could be renamed from default to shared (if using a final class instead of a struct).