Even if Chrome is making a system run "hot", the system should not overheat and shutdown if the laptop is otherwise working properly.
Everyone is focusing on the trees here, rather than the forest. Need to step back and take a look at the facts.
System is shutting down because it's getting too hot. This typically means that the system is not being cooled properly (not a huge surprise there considering it is a Mac.)
However, in the end, there is not enough data here to spot a definitive root cause for the issue.
Things that are missing, in no particular order (unless I missed it amongst all the "Chrome this", "Safari that" bits and pieces):
- Machine model and specs
- Was it purchased new or used?
- Any aftermarket mods or upgrades done to it (either by existing owner or previous if bought used?)
- Have any parts been replaced in it for any reason? If so, what was replaced.
Barring any of that info, here are some suggestions:
1. If it's an old system (like 3+ years) and it's used daily:
- the fans may need to be replaced.
- the thermal paste may need to be cleaned off and re-applied, and then allowed to cure properly before measuring new temps.
2. If it's a new system:
- thermal paste may need to be cleaned off and re-applied, and then allowed to cure properly before measuring new temps.
- (optional) have the fan(s) checked out. Possible that a defective one may be found, although more likely as the system ages.