How likely is it that
D1 forward voltage is slightly less than PNP Vbe
thus preventing the PNP from turning on?
Relay might have to be low current to do it quite that way, but rectifier voltage drop is consistently high relative to BJTs, particularly BJTs at low currents (~uA) so it's believable, or close. I would consider 1N4004 an intentional design element, not incidental, and would encourage testing it (including over temperature range..) before putting it down in production. A second BJT stage might be desirable, particularly if one only has rectifiers of lower Vf.
As it happens, rectifiers are made well over ratings; probably 99% or more of the time, if you see a datasheet where all parts in a series are specified with one or two classes of limits (typically Vf, Ir, Cj), they're using the same (max rating plus safety factor) part to cover each class. 1N4001-7 are specified in one class and you will see "1N4001"s break down at 1300V or so. When two classes are used, it's typically the <=600 and 800-1kV ratings, and the upper range typically breaks down at 1500-1800V.
This is partly no accident, as diffused junctions are pervasive (as you can more or less determine from the Cj(Vr) curve), and diffusion is notoriously difficult to control; but that alone doesn't explain the lack of low-voltage parts. It just happens that, making diffused rectifiers, and making them robust enough that people don't complain (probably?), and the thinner junction / lower built-in potential just doesn't save that much -- a hundred millivolts here or there. Hence, Vf doesn't vary much between parts, and clusters high relative to a signal BJT's Vbe.
Tim