As I am someone who works in an American Embassy I can provide some insight.

First as previously mentioned, just walking into an Embassy doesn't work, you will need to make an appointment. One reason is simply because of workload. Depending on the country the consular officers could be doing well over 100 visa interviews a day. The entire funding for Consular Affairs is based on fees for services provided (if you want to write to your congressperson to ask for money to be allocated from the federal budget so we can hire more people, we would certainly appreciate that). Call for an appointment, and normally time can be found rather quickly for an urgent situation for an American.

What can we do for the average American?

If you are destitute, the Embassy can provide you a loan so that you can go back to the United States. If you lose your passport, we can issue you an emergency one that will allow you to travel back to the United States. If you are arrested we can make sure you are well and advocate for the local justice system to treat you fairly (note we can not get you out of jail, you are in a sovereign country and you have to follow their laws).

All Embassy's have an emergency number that you can call and your call will be answered. Note just because you get a hold of someone at 2 AM doesn't mean they are going to do something at 2 AM, for many things nothing can be done until the next day. If there is a death, serious injury, or arrest of an American citizen, please do call us.

As to why you should register, I admit I don't register in western countries, but there are some places where it is very important to. In Nepal after the earthquake, the first thing I would imagine they did was go through their list of registered Americans and contact them to check if they were okay (and therefore able to provide that information to your family if they called the State Department to ask). If a country is going through unexpected upheaval and you can't leave by commercial means (note when we "evacuate" Embassies the vast majority of the time it is done through commercial transportation), that list would be use to see who needs help to depart the country.

Personally when random Americans get my contact details and want to chat, I generally do.

I will leave a link to this story of a Consular Officer that prevented two children from being sexually abused by their American citizen father and subsequently testified at his trial.