Still water is heaven for proliferating algae and bacteria.

A large city with large bodies of still water is a recipe for plague spreading. Having water on the move keeps the water clean and, as a consequence, the city healtier.

Also, waterfalls help enriching the water with oxygen, helping bacteria to decompose organic matter suspended in it, again improving cleanliness.

Low-tech climate control: the waterfalls fill the air with spray, which evaporates off and cools the populace.

The Romans used to achieve a similar effect with fountains.

Energy storage.

You use a form of energy generation with unreliable or periodic output. As a form of energy storage you pump water up into a nice big reservoir when you have surplus, then power turbines by letting the water spill back down to a lower reservoir to create a more steady power supply.

It might be more efficient in an enclosed system of pipes, but for aesthetic reasons and public morale it’s much nicer to have exposed waterfalls in public spaces. As @PluckedKiwi pointed out in the comments, the presence of additional waterfalls/ absence of usual waterfalls can act as a very public indicator of current energy supplies.

It’s pretty simple, pretty reliable and pretty.

There are several purposes which you can fulfill with a structure of channels and waterfalls.
On the one hand, channels within your city provide a fast lane means of transportation in any age of human technology. As L.Dutch pointed out, keeping the water moving instead of standing still keeps bacteria and algae from developing into serious health problems for your citizens. They can also provide either a source of fresh water or a sanitation possibility (more plausible in a low tech setting) for your citizens. Definitely, if have water channels and even more so, if you have water falls, you are provided with a natural way of micro-climate regulation. Cities tends to heat up in summer time/during heat periods, which water can compensate providing a cooling effect and keep the air from becoming too dry to make infections of the respiratory system more likely. In addition, the aesthetic effect of water channels and park lanes or avenues alongside it should not be underestimated - it keeps your citizens happy alongside more efficient infrastructural uses. The platforms above water falls can also make attractions for the city's more advantaged citizens/nobility or possibly every citizen and demonstrates the power of their ruler to be able to build such an amazing structure.
In addition, these large waterfall structure can actually have the turbines of your city-wide power plants at their bottoms without limiting the effect of any of the advantages listed above.

Waterfall is power

In Shannara Chronicles castle of Humans was built on what looks like hydroelectric dam:

Both having electricity (or other kind of energy if it's fantasy) and being able to dry or flood large terrain is pretty utilitary, and nothing shows it off better and is foolproof more than simple waterfall - it keeps maximum level of water in reservoir without any moving parts that could break.

Graduation towers

If you have a salt mine nearby, waterfall can be used to increase salt in solution, an important step in salt purification. Sure, most common design is based on wooden sticks, but regular waterfall requires far less maintenance.

refugees living around repurposed heavy industry, so my "waterfalls"
need to serve a utilitarian function (at one time if not currently).

Vital minerals were found under a large natural waterfall (Niagara etc.) In order to mine them, a method of diverting the water away was invented that didn't require changing the course of the whole river. That was impossible for large-scale geographical/geological reasons.

The old civilisation simply built chutes outwards from the top of the existing falls. Eventually the outward aspect of the mine became huge and more and extended chutes were added at different levels. People who lived at the mine workings had their own personal chute to keep their house or front door dry. This all proliferated in a haphazard and ad hoc way.

You're looking at it backwards. Nobody added artificial waterways to a city. They did what they had to do to support building a city on top of existing waterways.

Human settlements naturally spring up near water sources for a number of different reasons. An annoying drawback of these water sources is that they limit the settlement's ability to expand. When the settlement evolves into a city and you need more room, you can't simply move the water source out of the way (unless you're Dutch, of course). In your image, the cliffs add another limit to how far you can expand your city.

You're trying to make the most use of the land available so you've clearly built your city all the way out to the cliff edges and to the very banks of the river. Rivers are notoriously restless things, though. They meander all over the map and when they run over a cliff edge, they slowly erode away the cliff. Neither of these are particularly good things when you have roads and buildings in very close proximity.

A relatively cheap and easy solution is to constrain the river so that its course is fixed. Line the river channel with concrete to prevent meandering, and design outlets at the cliffs that prevent erosion. Add artificial channels with multiple outflow points to increase your water throughput, which helps protect your city against river flooding.

You'll end up with a large-scale erosion control project where you've carefully replaced the river's natural bed and channel with erosion-resistant material, and where you carefully monitor and control water levels, diverting water as necessary to avoid flooding. To an outside observer that didn't see your city before the project, it looks like you built a bunch of waterfalls for some reason.

If staged and planned right artificial waterfalls can increase the sound quality to an area. Flowing water can dampen sounds that are undesirable and create curtains of white noise to block out anything that could be considered undesirable as noise in an area. I've worked construction jobs before where if you had a river or waterfall near by you wouldn't hear as much construction depending on which side of the water you were on.

Your city gets its water from nearby mountains, however due to the proximity and difference in height the water arrives at very high pressure, too high for the primitive pipes of the time to handle. Since the city does not have valves and other control systems more importantly part of the city needs that high pressure (water wheels). The simplest way to bring the pressure back down so the water can be sent to other districts is a waterfall which releases all the pressure at once in effect resetting the pressure gradient and bringing it back down to a manageable level.

But the city is clever the pressure is used to drive a water wheel system, overshot water wheels in which the water comes from above are much more powerful than undershots ones. So they have row/stacks of water wheels powering various mills around the city, but you cant push all the water through wheels it is too unpredictable and will wreck tour mills. so excess water simply goes an alternate route and falls (see image). The city designer was smart enough to realize leaving the waterfall design allows for the city to build more mills and wheels later, allows the mill wheels to be well controlled, and lets the same water be sent to homes, irrigation, ect. You will see many falls staggering downward to generate different pressures for different purposes, high pressure for irrigation, low for drinking water, ect.

Niagara falls used to have a massive mill complex, but the mills could never use even a fraction of the water available there was just too much. later tourism became more profitable than the unsightly mills. If a city had been built around both the unsightly mills might never have disappeared.

Defensive purposes:

The city is on water, but cannot be attacked on water, because the the waterfall is making an almost unescapeable stream. The city is also granting very less option for an ground attack, because of such small bridges, which are even destroyable or which can be pulled in like castle bridges.

If the bridges are completely destroyed the city is only accessable to creatures(/machines) which can fly or can resist a waterfall stream.

That makes good military bases.

You could also work it into your story, by allowing the water to be exchanged for other substances substantially more dangerous, such as burning oil. Then it would not only be an aesthetic choice, but also a defensive choice.

Energy storage to maintain a study spot power supply

Hydathodes

https://www.botany.one/2017/07/hooray-for-the-hydathode/

Plants sometimes can build up internal water pressure. The water needs somewhere to go or it could burst the tissues. Some leaves have special pores called hydathodes. These pores release excess water in a process called guttation.

So too your city. Underneath there is excess water pressure. It must be released in a controlled fashion or it will invade buildings and ruin roads. The hydathode towers allow the pressure to be contained and released.

Bonus: Hydathode is a rocking fantasy name for a city.