I know, I watch too many Youtube videos. Random feed, lots of topics. I see more than my fair share of steam trains.
It seems like many steam operations always blow down when crossing a bridge. A bridge provides a safe spot for the hot blowdown steam/water to freely fall into the ravine or creek without any concern of scalding anyone.
I claim it is done purely for show.
I think it is an unneeded risk.

Blowing down the boiler can be done at the terminal before departing for the day, or at the end of the day after the last train. The engine is not moving, and there is adequate personnel and tools and time if something should go wrong.

Blowing down on a bridge, or when underway anywhere, is a risky situation. If something gets stuck in the valve (scale, staybolt remains, mud, etc) or the valve fails (lever, reach rod, valve stem, etc breakage), then the valve can be stuck open or partially open. This can cause an unscheduled stop for service, or at worst a dead locomotive if the valve remains stuck open.

Sure, if the water chemistry is such that the engine is foaming, priming, or otherwise NEEDS to be blown down, then do it. Otherwise, why blow down any time when running??

Continuous blowdown IS, in a sense, blowing down ALL the time when running. (Of course, like the 'self-cleaning' smokebox, this is no longer particularly tolerable 'ecologically', to say nothing of the various maintenance problems continuous blowdown might produce...)

In that respect, I have to wonder whether any boiler using 'proper' boiler treatment ought to be blown down over a State or Federally regulated waterway. Those plumes and rainbows sure are purty, but they also contain various materials requiring careful disposal details.

I confess I've always been in the camp of the 'don't blow down in plain sight of the onlookers' people. Particularly when your blowdown is directly on the side of the firebox, at 300psi...

On many Canadian National Steam locomotives, there was a sign above the engineer advising them to blow down the boiler every certain number of miles, with different values for freight and passenger service.

Some engines like CNR 6213 had pipes fitted to the blowdown valves that would vent from a muffler under the cab. Other engines, like CNR 6213, had the blowdown valves vent straight out the side of the engine.

I can't imagine that engineers were stopping to look for a safe place "to take a leak"...

Heck this was a time, when passenger car toilets drained straight down to the tracks...

As for Continuous Blowndown, with the design that CN used (from NALCO), the valve would be opened from a sensing line mounted to the cylinder blocks, and would went through a muffler/steam separator mounted either in front of, or behind the cab.

Then of course there was the Signal Foam Meter, which likewise vented through a similar steam separator/muffler.