Cleaning: My BB gets reefs of dried and hardened road debris building up. Its easier to see from the left, but removing the chain lets you wash that area better. I would also drop the rear wheel and clean the cassette in with the chain below. Once the wheel is out I'll inspect the rim brake track and the brake pads for embedded metal. And the front too.
I'd also scratch any dirt off the jockey wheels and rear derailleur, but I wouldn't unmount them unless they're not spinning well.
BB/Frame inspection: The chain makes it hard to get around the bottom bracket area, and the spider/chainrings also get in your way. With the chain off you can get in a bit easier to look for issues.
Chain clean, inspection, and wax: If the chain is off I will wipe it clean, then degrease in an ultrasonic cleaner. Then dry it naturally or using hot air. Then I'll deep fry it fully submerged in molten paraffin wax for 10 minutes or until it stops streaming bubbles. Then I fish the chain out with an old spoke wire hook, and let it hang and cool down. I refit it to the bike later and then sweep up all the little chips of dropped wax.
Reassembly: confirm the shifting's good and brakes are working effectively, and that there is no frame rub. Last thing is to inflate the tyres (always some bonus speed there)