You're saying you have a circuit, with a breaker, marked "family room". This circuit has more than 8 outlets. And then you had an event you call an "appliance short", and this did not trip the circuit breaker (!), and then after that, eight of the outlets do not work, while other outlets on the circuit continue to work. Ok.
Outlets are supplied from a breaker, usually in a string topology with wires or cable going from breaker to outlet 1, cable from outlet 1 to outlet 2, etc. etc.
However it is perfectly acceptable to have a tree-branch topology, with outlets 1-4 in a string, outlet 5-7 in a string coming off outlet 1, then outlets 8-10 string coming off outlet 3, and outlet 11 teed off outlet 6. This isn‘t super common since it's more work to wire this way, but sometimes the layout of the building demands it. Since you are not in the UK, loops are not allowed.
Both hot and neutral must go the same ways and follow the same paths. If you think of hot and neutral as two separate trees in parallel, they must be the exact same shape. You can't serve outlet 12‘s hot from outlet 10 and its neutral from outlet 11.
The good news is, everything should be accessible. It is mandated by Code that every junction be inside an accessible box, without disassembling any part of a building.
You start by making a map of the circuit, where it starts and which receptacles it connects to. Most of the time it's obvious. Sometimes you have to pop the cover off a box and see what's going on inside. As said, most of the time it's a plain string.
Once you know the layout, find the closest outlet which has failed to the circuit breaker. That is the first place you look. The next place is the next closer outlet. For instance in the complicated tree example above, if outlets 2-3-4-8-9-10 have failed, outlet 2 is the closest, so you start there and look at outlet 1 after that.
Failures in wires are rather unlikely unless you've been recently nailing things into the wall, and even that's a longshot. What's far more likely is a failure in a termination, and the usual culprit is the "backstab" connection method, where you jab a wire into a hole in the back of the receptacle, and a spring holds it.
It's quite possible that your appliance did indeed short out, and a weak backstab connection added a bunch of resistance, which limited current enough to keep the circuit breaker from instantaneous tripping. This turned it into a race between the circuit breaker's slow thermal-trip mechanism, and the outlet incinerating itself. The outlet won. A lot of houses burn down this way. Some electricians' approach is to never use backstabs, the NFPA's approach is to mandate arc-fault circuit breakers.
Perhaps your guy used the above methods to fix it. It's also possible he just ran some additional wiring to connect the dead half of the string back into the same circuit. That is OK if he made it a tree structure, by identifying and positively severing both hot and neutral at the failure point. You cannot rely on the failed wire to be the severing, because both wires must be cut in the same place to break the loop. One wire looping and the other not is an even worse situation!