Doppler for pipes

Perhaps an ultrasonic flow meter? Plenty to choose from:

Things can get messy with diffraction through multi layer pipes, partially filled pipes (air and water), varying temperatures, changing fluid composition without tweaking the settings in the meter, etc. Note that there are different types (Doppler and transit time) for different use cases:

Saturated steam (with water droplets) will work with ultrasonic flow meters. Dry steam (no water droplets) will be a problem.

Hint: The reason that Kill-a-Watt meters are so popular is that the AC wall plug is standardized and found everywhere. That's not the case with fluid flow, which can have a wide variety of fluid types, fittings, plumbing etc. The most common fluid pipe and fitting is probably a 3/4" garden hose and 3/4"-11.5 NH fitting.

Those are some pricey meters.

I've seen ultrasonic meters used for sanitary sewer flow rates, but usually they're mounted above a notched weir and actually just measure the height.

Well, if you want cheap, there are paddle wheel flow meters

and turbine flow meters: The problem is these require the paddle wheel or turbine to be inserted inside the pipe, where they also partly obstruct the fluid flow. However, they are cheaper than ultrasonic.

Accurate, reliable, cheap. Pick two.

They also use weirs to somewhat limit clogs from FOG (fat, oil and grease) and fatberg obstructions to easily accessible locations. You really don't want to deal with a fatberg:

It's been a while since I was hands on in any mechanical rooms. But I vaguely recall we measured flow by pressure differential across an orifice. A flow meter was used for something like make-up water to a boiler, where accuracy wasn't critical and flow volume small. But for a boiler or cooling loop the pressure differential was reliable and cheap. And with 400 or so boiler rooms cost was important.

Welcome to fluid dynamics and Bernoulli's equation:

Works for any kind of fluid flow (air, gasses, water, sewage, etc).

However, there's a problem with using pressure difference to measure fluid flow rate. The plumbing and sensors needed to measure the changes in pressure caused by changes in fluid flow must be designed and built into the machinery in advance. In other words, one must plan and design ahead. That's where ultrasonic flow meters are best. They can be retrofitted into almost any machine by attaching the transducers to the outside of the plumbing. No need to drill holes, insert obstructions, pitot tubes or deal with internal mechanisms (paddle wheels and turbines). Ultrasonic sensors can be temporary or permanent as needed. If you're measuring flow rate in an open channel, adding additional hardware is not a major problem. However, inside a sealed pipe, ultrasonic is best.

400 boiler rooms in one location or scattered across the planet?

Drivel: I hate plumbing.

Military base. Not that much different from a college campus - barracks instead of dormitories, DFACs instead of cafeterias, lots of admin buildings and classrooms, maintenance shops, etc. Most of it was ancient when I started, about half rebuilt or new by the time my 30 years was up. Interesting job as standards and technology changed through the years.