There’s a difference between a safety inspection, which merely ensures you’ll get the sticker, and a mechanical inspection which checks out far more than the state requires. A car might pass the state’s safety inspection but still have major problems.
if you trust the seller to reveal problems and he says there are none, just go get your sticker and pay the standard fee. Don’t ask for any further work. If you suspect the seller might be concealing information, get a full inspection.