It's because world governments and companies spent a shit load of time and money changing things so it didn't become an issue. People point to it as a load of fuss over nothing, the only reason for that was that action was actually taken.
When I heard about the Y2K panic in the mid 90s the very first thing I did (as a kid) was advance my clock to 12/31/1999 to see what would happen next.
IIRC when the date changed, it went back to some date in the 1980s I think. This was either windows 95 or 98, I can't remember.
Anyway that advice to turn your computer off... What the heck? Then what, throw it away?
We told people to turn them off because it was just easier than trying to explain to them that they might have to restart individual programs if they bugged out. It was a simple way for people that didn't understand computers to get everything back into a known state.
Nothing, really. The worst that could have happened to personal computers was that they'd show a wrong date. What people were really worried about was how computers responsible for critical infrastructure would handle it.