It's been ages since I bought D-cells, but don't some of the major alkaline manufacturers put a date code on them like they do other sizes, so you're assured of how old they are? I want to say they rate for 10 years or something so subtract 10 from the best by date to know the manufacture date.
The linked motor is always under load due to the vibrating weight, and probably would wear itself out in months if not less due to the tiny sleeve bushing handling the stress of that, if not the brushes failing first, also considering no manufacturer brand is specified so unknown quality.
Consider the application it is designed for, momentary use not continuous duty where a small enough size to fit in a phone is more important than running hours lifespan, and it specs the minimum voltage as 1.5V so a single D cell is not necessarily suitable since it will be below 1.5V for the majority of its capacity... if it operated reliably below 1.5V, you'd think that they would specify that.
It could work fine for a while, but I'dwant to test rather than assume it, using a variable voltage supply to see how low it can go and keep spinning or if it will be stopped and started, how low before it can't spin up. I'd sooner use 2 x AA NiMH cells in series, but with the higher battery voltage and RPM, the motor would wear out that much faster.