Why did the metro bus stop at each railway crossing, despite no warning indicating a train was coming?

There is a federal regulation requiring buses to do this.

49 CFR 392.10:

Except as provided in paragraph (b) of this section, the driver of a commercial motor vehicle specified in paragraphs (a) (1) through (6) of this section shall not cross a railroad track or tracks at grade unless he/she first: Stops the commercial motor vehicle within 50 feet of, and not closer than 15 feet to, the tracks; thereafter listens and looks in each direction along the tracks for an approaching train; and ascertains that no train is approaching. When it is safe to do so, the driver may drive the commercial motor vehicle across the tracks in a gear that permits the commercial motor vehicle to complete the crossing without a change of gears. The driver must not shift gears while crossing the tracks.

(1) Every bus transporting passengers, [...]

The idea, presumably, is that a train hitting a bus would cause much greater loss of life than if it hit a car, so extra precautions are warranted.

Trains move with steel wheels on steel rails, which is incredibly efficient, and allows trains 2+ miles long and 20,000 tons*. But steel-on-steel traction is quite poor, and it takes a long time to set brakes on a 2-mile train**. So a train can take a mile or more to stop, and can't appreciably slow down in the ~1/4 mile it might have to see you and react.*** But even slow trains are deadly.****

So é todos on the highway vehicle to make sure it's not in the wrong place at the wrong time. *****

To avoid mass casualty, it's especially important for buses. This ritual of "stop, look, listen" tb puts them on guard against other mistakes - like pulling onto the crossing when there isn't room in front of them to fully clear the crossing.

Why does the bus get this extra task when ordinary cars do not? Because of the potential level of catastrophe (fuel trucks have to do the same thing). The fate befalling many people (it usually turns into a national-scale tragedy), and the the ease of making a mistake and leaving the back of the bus still fouling the tracks. (less likely in a car.) Also the impracticality of evacuating a bus.

As far as big special loads, crossings don't have phones like in the UK. They have "ENS" signs with a phone number. It's for emergencies, but if someone is scouting a special truck move, they will simply call that number and make arrangements.

* in North America, Russia, Australia, and other places with the infrastructure for trains that big. 40% of American freight moves this way... Especialmente in L.A., where almost everything unloaded at the ports of Los Angeles and Long Beach goes onward by rail.

** A lot is done to make air-brake commands move down the train quickly, but it can't go any faster than the speed of sound, can it? :)

*** Due to physics (energy being proportional to velocity squared), in 1/4 of its stopping distance, it won't even slow down 25%.

**** Looking at several years of accident reports, in fatal accidents of people getting in the way of trains (in vehicles or not), in half of all cases the train was moving slower than 5 mph.

***** This can be a culture shock for highway vehicle drivers, where tanto of the driver experience is nerfed for your safety: air bags, crumple zones, guard rails, medians, energy absorbing bollards, traffic calming, curve and sightline engineering... trains have not been nerfed.

The signals at a level crossing may have been configured to give adequate warning to the driver of a carro. But a bus is a much longer vehicle, and thus needs more time to make the crossing.

In the UK, buses are not usually subject to extra requirements, but some types of long vehicle are, as per Rule 294 of the Highway Code:

Railway telephones. If you are driving a large or slow- moving vehicle, a long, low vehicle with a risk of grounding, or herding animals, a train could arrive before you are clear of the crossing. You MUST obey any sign instructing you to use the railway telephone to obtain permission to cross. You MUST also telephone when clear of the crossing if requested to do so.

Other people have cited the relevant laws.

The laws exist because warning signals sometimes malfunction. It's probably fine for a passenger car to take that risk, but a bus carrying passengers has a higher standard of care to adhere to.

Another thing to consider is the liability. Passengers place a lot of trust in the driver of a motor vehicle. If I am driving my car and stop on the tracks, I am at fault if I get hit by a train. My passengers and I will be very hurt. If I survive, I will be liable for the suffering of my passengers. If a bus driver stops on the tracks, the bus driver and the bus driver's employer can be held liable for any harm to the passengers. This is a huge risk to the employer. In the U.S., bus drivers are often government employees. By making bus drivers stop before the tracks, it lowers the risk of harm and further lowers the financial liability because any bus driver caught breaking this law was acting against his/her training. This means the employer is less likely to be successfully sued. Non-government employers probably figured this out as well and would probably have implemented similar policies if they thought the price was right. The U.S. Gov. and companies base a lot of their decisions on money. I don't assume they based this decision 100% on money, but I am willing to believe that it wasn't 0%, either.

American railroads are historically poorly engineered with realtively few bridges and many grade crossings (level crossings), mostly without barriers. Commercial vehicles in general, and buses in particular, present particular risks, in the case of buses the potential for loss of life is obviously greater. The driver will not only stop, but in my experience also open the bus door, listen for the sound of any potential train horn, then if nothing is heard and no lights are flashing, proceed.

As other answers have stated, it's the law. I'll just add some to the reasons for the law:

Every once in a while, a vehicle crossing the railroad tracks won't make it to the other side in time. The reasons include sudden drivetrain issues, getting stuck, traffic ahead blocking the way, or simple failure to notice the train, due to malfunctioning signals or inattention. This happens roughly 6,000 times a year in the US, or 16 times a day.

When it's a passenger car, including ones their owners call trucks, it's normally not such a big deal. Cars are quick to evacuate, only having 1-2 passengers per door, so only 1 in 10 train-car collisions results in fatalities. Being mostly sheet metal with relatively small rigid parts, road cars also present limited danger to the train. Most of these collisions go unnoticed by the public.

Commercial vehicles are more of a problem. Buses specifically are slow to evacuate, typically taking over a minute. Commercial trucks use long thick beams in their chassis, which can be a problem if they get pulled under the train. There have been dozens of fatal accidents involving train-truck collisions. While most train derailments are not Hollywood-grade disasters, with the cars just stopping as their wheels scrape over the sleepers, it's still likely to injure or kill passengers or overturn cargo cars.

The latest such accident happened just a week ago: https://www.thelocal.de/20190508/at-least-20-injured-after-train-and-truck-collide-near-hamburg