Surely a normal instrument scan would have picked up that something was wrong (nose up with speed increasing, altitude unwinding, VSI indicating a descent... hmmmm...). Then once you've been clued in to a problem you can check other instruments. (I wonder what the FO's ADI looks like? Hmmm... shows nose down. What does the standby look like? Hmmm... it shows nose down as well.)
That process shouldn't take very long at all, and probably should've happened within seconds of the initial climb showing on the faulty ADI. In short, I don't think routine monitoring of the standby is necessary. What is necessary is having a good overall picture of what the aircraft is doing, in other words, one mustn't allow one's self to be a numpty in charge of a high performance jet.