Actually, a piston aircraft could compete with a jet aircraft, but not in terms of speed but in maneuverability. A P-51 could out maneuver any modern day fighter jet.
Given how this is kind of buried now I was just going to forget this but...I can't.
Yes, you're correct, but it's also a lot like saying "a '66 VW bug has a tighter turning radius than a Camaro ZL1 that's just leaving the assembly line." Let's go over what's wrong with that:
1. The piston aircraft is more maneuverable than a jet aircraft by virtue of the physics of speed. The faster something moves, the larger its turning radius. That said, speed is a heck of a lot more valuable to a tactical (or strategic - the B-52 and the B-2 are both amongst the fastest subsonic aircraft flying today) combat aircraft than maneuverability. That's why they make such a big deal over the F-15, F-22 and Sukhios being so fast. Already by 1944 if not earlier (certainly in the middle of WWII at the latest) they started realizing this. The P-47 and P-51 weren't as maneuverable (depending on how you define that) compared to its adversaries, especially the Japanese, but they were fast (especially the P-47 at higher altitudes). P-47 and P-51 pilots could often engage, disengage or ignore furballs at will. From a purely tactical perspective the "engage at will" part was especially important as most kills were made on the first pass - and if the American pilot failed that, often he could simply disengage and move on to something else.
2. Maneuverability is more than just simply having a tighter turn radius. I'm going strictly off memory, but certainly the Bf-109G could outperform the P-51 in certain areas (I think it had a better roll rate) but the P-51 was more maneuverable in other areas (IIRC the -109 had superior low-altitude performance but the P-51 had superior medium-altitude performance). At low-medium altitude the P-47 was a dog but at high altitude it could maneuver period where other aircraft struggled to do more than go in a straight line (if even that). Depending on how you define "maneuverability" the latest Su-35 is much, much more maneuverable than the P-51, doing things that are physically impossible on the ancient 40s-era frame even if you took it far, far beyond structural limits.
3. Maneuverability isn't even that important anymore. There's a common fallacy that I've noticed that boils down to "once a lesson is learned, that lesson cannot be changed or considered obsolete and is immutable for the entirety of what's left of human existence." Way back when, we're talking centuries ago, there was this musical instrument called a harpsichord. You probably don't even know what that is or what it even looks like. That's because sometime in the 1600-1700s it was replaced by this thing called the "piano" which could do everything the harpsichord can - and then some. That said a lot of people were skeptical and it took a while for pianos to take hold - but now the harpsichord is virtually an extinct technology.
You probably know what a typewriter is. In the 70s they came out with "electronic typewriters" - they are exactly as they sound, originally simply electronically-powered typewriters that printed on a dot matrix and then later featuring CRT screens, essentially an extremely primitive desktop computer with word processing ability only. These machines had an extreme difficulty in holding any market value whatsoever and in many cases disappeared because, well, they just sucked that bad. The screens usually displayed that green-on-black font that's associated with ancient DOS machines (not the most conductive for a word processor) and required expensive, heavy and awkward equipment to match the print fidelity of even a stinkin' typewriter - which usually was an actually typewriter essentially attached to your machine. Otherwise you're stuck with a dot matrix which, well, you can just forget about high-fidelity printing at that point (pretty much useless for commercial printing and was pretty much for internal documentation only. The U.S. Justice system even refused to recognize anything printed as a dot matrix as a valid legal document).
Then in the early 90s IBM and especially Apple came out with computers (and just as if not more importantly - Microsoft and Apple came out with the right software) that could not only print with equal fidelity than typewriters, but in fact
better, and with literally every font on Earth (and even facilitating the creation of new ones) without having to switch out entire machines. Now the only reason why you'd ever want to use a typewriter is if you have some sort of weird nostalgia fetish and it also became a nearly extinct technology.
So what's the point of that long-winded pseudo-history lesson? The important lesson is that people were shy to adopt pianos because they were skeptical. People laughed at early electronic typewriters because they really, really sucked. The technology hadn't proved itself yet, or was far too immature because people rushed in to adopt the technology before they had any data or technical ability to refine it (and I should note that this is a necessary step - it's how you get the data and experience to refine that technology in the first place. Sitting on a word processor for 20 years because you haven't perfected ink jets yet is really, really stupid). But eventually, the technology caught up and blew away whatever came before, and now people laugh at you for using obsolete technology. You need to keep that in mind for the next thing I'm going to talk about.
So here we are in the 50s and 60s when the United States Navy was trying to perfect a technology the Germans had worked on in WWII in order to allow interceptors to better shoot down bombers. The end result was the AAM-4, later re-designated AIM-7 Sparrow. Meanwhile the Air Force had their own program, the AIM-47 Falcon (the reason for the wide disparity in numbering is due to the fact that you have a lot of ballistic and cruise missiles taking up the middle numbers). These were really great weapons during the time they were conceived - you can stand far beyond the active defense capability of a bomber (to the point where active defenses for bombers became pointless and a waste of space) and just sit there and let the missile do all the work for you. That's a whole lot better than getting into gun range (remember, the Tu-16 and Tu-95 - and the B-36 initially, mind you - were bristling with 20-23mm autocannon) and trying to actually aim at stuff. But in Vietnam it was shown that the technology was still immature and had built-in limitations due to unanticipated combat conditions - missiles needed to do more than fly in a straight line at bombers flying equally straight lines. Fighters still needed the proverbial knife to back-up their guns in a gun fight (or a better analogy, a bayonet) because missiles were still at the stage were guns needed to be reloaded by the muzzle and part of that reloading procedure involved tearing a paper cartridge with your teeth.
Well, just like you never really see bayonets anymore (or harpsichords, or typewriters) missiles are now improved to the point where a backup isn't really necessary. AIM-7 Sparrow got improved to the point where it far out-numbered gun kills in Desert Storm (and keep in mind there were a number of Sparrow kills in Vietnam too, once people learned to work around the limitations). AMRAAM and Meteor make even these improved versions of Sparrow look like a joke. We now have missiles that can fly
backwards relative to the launching aircraft. Python 4 and 5 were the first missiles to really showcase this technology but AIM-9X and ASRAAM can do this too. Now people are seriously wondering if a gun is just going to be an expensive waste of space, and if designing a plane to be maneuverable is just throwing money at an un-called for capability (DAS on the F-35 will allegedly give it 360 degree engagement capability while flying in a perfectly straight line). Of course, DAS is an immature technology....