I think that the answer to the question arises more from a 'pure logic' analysis than from an airmanship (aviation specific) analysis.
If you knew ahead of time that you were going to have an engine failure after takeoff - for example, 30 seconds, 60 seconds, or 5 minutes after takeoff - you would want to maximize the amount of energy you had in reserve prior to the engine failing. In an aircraft, we have two types of energy, kinetic (forward speed) and potential (altitude below us). If the engine is working, it provides a third source of energy.
It appears to me that climbing at best rate (Vy) would allow you to put more energy in the bank prior to your engine failure than climbing at Vx.