Surely it would get more thrust from a three blade variable pitch propeller. From what I've read it's all about the area swept and also should be far enough apart to not cause too much interference. Besides having a constant speed prop would allow for a wider range of optimum operation.
You're not wrong, but a fixed pitch two-blade prop has a pretty big advantage: it's cheap. Cheap to buy, cheap to install, cheap to maintain, cheap to repair. 172s are predominantly trainers in FBO fleets, and cheap is the winner. The efficiencies of a better prop are lost when you're just doing pattern work and short cross countries.
Also, when you're not climbing, a two-bladed prop is actually going to be more efficient, not less as it's creating less drag. According to the linked Hartzell FAQ:
Depending on the combination of [engine power, operating RPM for the propeller, diameter limitations, design requirements, etc] a 2-blade propeller may be most efficient, but as power increases additional blades are generally required to efficiently utilize the increased power.
This means that on a 160 horsepower engine like the 172 is rocking, you aren't going to see particularly tangible benefits from increasing the number of blades. Going to a constant speed prop would, on the other hand, give you greater utilization of power across the entire flight envelope.
An overhauled McCauley fixed pitch 2-blade propeller that fits a Cessna 172 is $2,100 on ebay. It has no moving parts (I mean, aside from itself) and is relatively easy to balance.
A (non-overhauled?) Hartzell 3-blade constant speed prop off a Piper PA-46-350 Malibu Mirage is $15,000 on ebay with no logs, which is usually a bad sign. It's not a direct comparison to, say, the prop on a C182, but it's probably in the ballpark. It's harder to balance, and requires more maintenance to make sure the constant speed mechanism is up to snuff.
I was only able to find prices on ebay, which isn't the most scientific comparison ever, but it's a start.
I can't improve on @egid's answer on efficiency or cost, but another point on constant speed units is related to 172s predominantly being trainers: fixed pitch props are easier to fly.
My first few lessons in the circuit were busy enough just getting the altitudes, positioning, radio calls and checks completed correctly. When I did differences training for variable pitch a few years later it suddenly became busy again, simply from trying to remember what the RPM and MP settings I was supposed to be using at each stage (generally 5 different combinations per circuit).
It's not much harder, admittedly, but just having one dimension (the black throttle knob goes forward for more power or back for less power) was easier on my already over-taxed student brain than the two-dimensional variable pitch setup (black knob for power, blue for RPM; with blue going forward both when you want more power (takeoff) and when you're decreasing power (short final, in case you need to go-around)).
For a given horsepower two blades are more efficient than three, three more than four, and so on.
There are a few reasons to have more than two blades:
-the engine generates so much power that a two blade prop would need to be so long that it would contact the ground.
-the engine generates so much power that a two blade prop would need to be so long that the velocity of the blade tips would exceed the speed of sound.
-the application of the aircraft requires acceleration and climbing ability over top speed. More blades for a given power equate to slower top speeds but better takeoff acceleration and rates of climb.
-in the case of the V-tail Bonanza, the added weight of the three blade moves the CG forward making loading easier.
The most efficient prop for a given hp is a single blade (it exists). Plenty of examples in Google images.
The cost and manufacturing/maintenance aspect are other considerations and have been addressed in other answers.
This is not a hypothetical question since Cessna did manufacture versions of the 172 that included constant speed props. These included:
- The FR172, aka "The Reims Rocket", produced in France from the late 1960s to the mid 1970s.
- The R172K Hawk XP, produced in both France and the US from 1977 to 1981.
- The Cessna Cutlass 172RG, produced from 1980 through 1986.
However, none of these aircraft proved especially popular with the general public. Even with higher horsepower engines, the common perception was that the increases in speed and useful load were too small to justify the additional cost and complexity.
When Cessna started producing propeller aircraft again in 1996, they focused on their most popular models, such as the standard Cessna 172 and the 182.
If you wanted an an aircraft with significantly better performance than the 172, you were expected to buy the Cessna 182, an aircraft which was very popular with the public. The Cessna 182T has a 3-bladed constant speed prop.
So, to answer your question, yes those kinds of improvements would have and did increase the performance of the Cessna 172. But they were not popular and, thus, were not worth the effort.