The extinguisher isn't required.

It can be useful, but really needs to be qualified.

Let me ask this: have you ever fought a fire? Have you ever fought a fire in an aircraft? Have you ever had a fire in a small aircraft when you're flying (as opposed to someone else flying)? I'm not asking to be facetious.

I'm a long time firefighter; structural, wildland and in the cockpit. I've had a few fires on the flight deck over the years: electrical, hydraulic, fuel, etc. I've had hydraulic motors leak then catch fire, wiring catch fire, even janitrol heaters dump fuel and burn. I've also had an extinguisher discharge unintentionally twice in a cockpit, both times dry chemical. I've been around others who had fires and once watched a buddy make a quick return to a runway in northern Iraq after a very rapid electrical fire shortly after a night takeoff filled the cockpit with smoke and burned up a lot of the interior. He made it. In each of those cases, there was never an opportunity or occasion to use an extinguisher.

Certainly there are times, but in my limited experience, when you're on fire, especially if your'e flying, especially if you're in a small cockpit and light airplane, the chances of you actually using the extinguisher aren't great, nor is the opportunity. If the pin gets pulled, however, and the extinguisher discharged (flight suit pant leg caught on one once, and a passenger stepped on one another time), it makes a choking mess in the cockpit and can look a lot like a fire.

I think an extinguisher has value, though in that 172 the value may be more for a fire outside the cockpit than in. A dry chemical extinguisher in particular isn't going to do a lot for an electrical fire until you get the electricity shut off and remove the ignition source. Beyond that, there's not a lot to burn, and you'd best be getting to the ground.

I had a cockpit full of smoke and no known source a number of years ago; I couldn't see well as I crossed the ruwnay threshold and couldn't see or breathe as I rolled out. In that case a hydraulic dual pump had run away, overheated, burned up the seals, and pumped out H5606 fluid, which subsequently caught fire. I did use an extinguisher, but only after I stopped and shut down, got out, removed the panel covering the hydraulic pump, and shot the fire.

On another occasion, following a forced landing following an engine failure, heavy black smoke was coming from the engine intake. I was located on a mountainside with tall grass, fuels, slash piles and wind gusting 40 knots, and downwind from a large and very active wildfire. I pulled the pin on the extinguisher, circled the aircraft looking for an ignition source or fuel leak, and then attempted to climb out across the nacelle to discharge the extinguisher up the intake (nowhere else to apply it and no other options). I removed a nomex glove, put my hand on the nacelle and burned my hand, and decided not to lay across the nacelle: thus, the extinguisher wasn't used. The smoke abated, as it turned out to be what was left of the oil cooking off, and the aircraft remained intact. In fact it was used again.

The point is that while my own experience is limited, I haven't seen that many occasions when discharging the extinguisher in the cockpit. helped or would have helped, or would have been possible. I've put out fires with my jacket (nomex jacket on a burning janitrol) and other things, and have stopped fires by removing the ignition source (cut off electrical, etc), but there may be a bit more to the issue than you think, and it may be that the owner of the aircraft has had use and abuse of extinguishers to the point that he's removed them.

If you're uncomfortable with the arrangement, you might carry your own (for which you become liable, and which if not properly restrained could become dangerous). I'd certainly consider it. You might consider going elsewhere. I don't know that the extinguisher points to systemic problems with the operator: you'd have to look more closely (and should if you're flying there, or considering flying there). It may or may not be a good match.

If you've never discharged a dry chemical extinguisher, especially if you've never done it in a tight, enclosed space, take my word for it that you'll be tasting it for three days after and it will still be in your hair, ears, and other places that seem impossible to reach. It also goes into a lot of places that get very expensive, such as radios, instruments, etc, and forms a sludge and cake in contact with fluids that can take a long, long time to clean up, if ever. In engines, it can ruin the engine. It may be the only option, but it's also not a great option. Conversely, Halon is hard to get, expensive, and displaces oxygen, while reacting with fire to form phosgene gas, among other things, and it works in an enclosed space by keeping it enclosed...meaning that if you ventilate after use you lose most of the utility of the halon (you must vent if you plan to survive). Then there's CO2, which in an enclosed space is a suffocation hazard. For interior fires...water, though it's not seen very much. A water bottle, on the other hand, does nicely. Then again, so does shaving cream, believe it or not (subject for another thread, perhaps).

You shouldn't get on an aircraft with a parachute and not know and understand the parachute (should have used one before). Same for fire extinguishers; know them, know their limitations and liabilities. They can make your day worse, too. Flights have crashed worrying about minutae when they should have been focused on task #1: fly the airplane. I've often carried my own extinguisher in light airplanes, but on the basis of knowing what I could and couldn't expect from it, and how it might be employed, and what it might mean in the wrong circumstance. Fire tends to incite the flight portion of the fight or flight response; people tend to get worked up over fire. Remember the same thing it says by the swimming pool: walk, don't run. We can do a whole thread about that, too. Just remember to consider every side of the coin, not just the one you see when you open the door.