debugging is one (perhaps the only/primary) need. if you have stuff only breaking on production systems, it may be because of a number of factors that aren't easily replicatable in a testing environment. Sometimes the only feasible way (assuming you want fast answers because of live production downtime) is to get on to production servers and look at things there. In most cases read-only access should be fine.

You can't reproduce a dev/staging environment with the same traffic pattern, user behavior, and request load as to a production environment. Google bots won't crawl your dev/staging environment and you won't know its impact until you see it. DDOS attack won't happen in your dev/staging environment. Click fault won't happen on dev/staging.

Developers probably shouldn't have root privilege on production but they need to have read access at the very least. They also should have write and deployment privilege on a few production machines for testing and experiment. Whatever mess they made there should be counted under acceptable loss.

No disrespect, but after a statement like this, "...deployment privilege on a few production machines for testing and experiment.", I wouldn't let you near my servers. That's not what a production server is for.

Ideally, this is true.

But, I've worked in places that had bad resource planning, and didn't allocate the same kinds of resources in staging. When we had performance problems, there was no place else to go.

It can be hard to convince people to take their expensive hardware and IT time to set that production system up, and now, "duplicate it so we can debug". Eventually, I think you just give up and go somewhere else.