Whilst I understand your rather-be-safe-than-sorry approach and acknowledge that it is the best advice to give to a beginner who's chain you cannot inspect, the reality is slightly different. A chain damaged like that (side-plate bent and popped off the rivet) can only be damaged across one link. Make that half a link if we want to be technically correct. Repairing or replacing that link does not leave you with half-broken links elsewhere unless you keep on doing the same mistake over and over again. The mistake has to be rectified otherwise it will happen with the new chain as well.
Links break like this at a very specific point in the link's travels around the drivetrain. The way a shifter works is to, at the bottom of the cassette where the chain I is slack, bend the chain so that one link is on one sprocket and the immediate link after that is on the next sprocket. As that crossover travels from slack (bottom) to tension (top), it is vulnerable at the 5-to-twelve position where the transition between slack and tension is immediate. At that point there mustn't be tension on the chain and most of us have learnt to instinctively ease off when shifting. But, should you not ease off, only that transitional link gets broken, nothing else. We know this because no other part of the chain is bent at that stage. Further, the quick nature of the failure prevents this for occurring half-way across many links. The reason for this is that the chain either breaks completely or the leading edge of the outer plate separates from the pin, gapes open and then gets caught in the jockey cage at the back as it travels backwards.
There is no opportunity for multiple half-broken links.
