Well first off challenge your perception that you can't use a conventional wood screw in particleboard/chipboard. You can. Screws (of a number of different types) are available that are designed specifically for this purpose, and have been since at least the late 60s.
Even when there were no such alternative screws (or the budget didn't allow for them) old-style normal woodscrews were used, possibly with some workarounds one of which is pictured in the previous Q&A you link to in the Question, a cross-grain hardwood dowel. Another method was to drill an oversize hole and glue in a fibre wallplug that the screw would subsequently be driven into once the glue had set1.
When are barrel nuts + machine screw to prefer over wood screw + hole?
When you want to build knock-down (KD) furniture. That's the primary purpose of barrel nuts.
And when are screw + hole better?
Permanent assembly, often but not always with some or all joints being glued as well.
I can also imagine that barrel nuts have a bit more tolerance if there are many wholes that have to line up, since they can move a bit in their hole
Not really. Although of course you could drill the holes for the nuts oversize to provide some wiggle room if you chose to, that's not normally how barrel nuts are done — the normal movement they have in their holes is really confined to rotation, with no or almost no side-to-side movement2.
while the receiving hole for a wood screw must be very precise.
The 'receiving hole', the pilot hole, for a wood screw can be drilled with the pieces temporarily assembled, meaning that alignment is automatically ensured; there's no additional difficulty here.
And these days various self-tapping screws are available that can be used on particleboard so that you don't even need to drill a pilot in the receiving panel, just the clearance hole (along with its corresponding countersink or counterbore of course) in the piece that will join to it.
1 This same trick was often recommended for screws going into end grain in DIY furniture plans in the past, because of the belief (whether accurate or not, see more on that here) that screws didn't provide sufficient hold in end grain.
2 IME it's normal for there to be zero play. How you build in some slack would be to drill slightly oversize holes for the bolt instead.