Just to put what I have researched:
I personally haven't researched far enough in history in regards to wrestling, but if modern and semi-modern practitioners are to be considered (particularly Ad Santel, Farmer Burns, Frank Gotch, Karl Gotch, Billy Riley, and the person who I consider the deserving inheritor of the art, Tony Cecchine), I always did believe Catch Wrestling to be the longest running, unbroken lineage of various forms of wrestling, ranging from Greek and Roman (aka Greco-Roman, and to an extent, Pankration) all the way to nearly all European and some Asian forms of wrestling (English, Welsh, Irish, Indian, Iranian, etc.). The lineage was preserved and continually practiced during the 1800s all the way to the early 1900s in traveling circuses and carnivals, where people has the chance to fight the best practitioners of Catch Wrestling for a large money prize, and if they defeat them (or perhaps last long enough against them), they'd win a prize (and if they lose... well, if they either lose via getting pinned for 3 seconds, giving up, getting choked out or getting their limbs broken, and they get no money prize))
There's an interesting take on the recent history of catch wrestling and how it may have influenced modern grappling martial arts like Greco-Roman Wrestling, Freestyle Wrestling, and even Brazilian Jiu Jitsu and Judo.
As I read before, Ad Santel may have introduced some dangerous submission techniques to Kodokan Jiujitsu (what we now know as today's "Judo") back then when he beat nearly every one of their elite Kodokan jiujitsuka became the 11-time World Judo Champion (or at least, proclaim himself as one seeing as no elite judoka could beat him).
I've read somewhere that due to the judokas being defeated by Ad Santel, Japan had an increasing interest for these techniques called "hooking" (which are what we now call "submission" or "submission holds", but "hooking" may be more dangerous due to its intent to cripple and maim, not make people "submit", especially considering Catch Wrestling love for finger and wrist manipulations while applying hooks, unlike that of BJJ, which uses concession holds). It may not be a surprise if Ad Santel defeated the judokas by outwrestling them then applying hooks like the double wristlock -- AKA the "kimura" submission (Ude Garami), which then led to kodokan jiujitsu attempting to integrate such "hooks" into their curriculum and make them safer for practice by in the process of turning judo into a sport in the modern era. That, and sending out judokas to various parts of the world in order to not only promote their art, but to also train with wrestlers in order to improve judo. One of such people sent out may have been Mitsuyo Maeda, AKA Konde Koma, the man who introduced Jiujitsu to the Gracie family in Brazil.
During the late 1800s, as it approaches the 1900s, Greco-Roman wrestling was introduced, and it took out the "hooks"/submission holds in favor of the core wrestling and pinning the opponent to the ground in order to make it safe as an Olympic sport. This is the birth of what we now call "Greco-Roman Wrestling" or "Amateur Wrestling".
Catch wrestling gradually lost its popularity during the 1900s (just a personal theory, but it may probably be during World War I, as most able-bodied individuals, like practitioners of various western martial arts, may have perished) all the way to the modern era, perhaps due other combat sports (particularly boxing) starting to gain prominence for being more exciting to watch than a grappling match (just a personal theory). Due to the losing popularity of legitimate catch wrestling matches, some catch wrestler practitioners, in order to make a living, created what we now know as "Professional Wrestling", a scripted fight using safe submission moves (aka "show holds"), showmanship, and exciting theatre-esque performance.
As I analyze it, in modern day, Japan still maintained the deadly submission holds, since their "show wrestling" still uses dangerous techniques like the "juji-gatame" (aka armbar... also may have come from catch wrestling as well), but they lost the core wrestling (ie the greco-roman aspect) in favor of submission holds, while in the western shores, due to the run for gold in the Olympics, we have retained the greco-roman aspect of wrestling, but we got our submissions back when BJJ came to the US. The connection from catch wrestling all the way to today's mixed martial arts are a long history, but very interesting as in all respects, old catch wrestling and pankration fights are a lot like modern MMA.
In any case, a Catch Wrestler does not want to have his back on the ground, not only because he may lose via pin, but it's also because in historical battlefields, being on the ground for around 3 seconds would usually mean death, as that's enough time for someone to stab and kill the grounded individual, so they prefer it if they're the ones that takes the opponent to the ground and hold them down there. This is also true in Japanese jiujitsu, where they emphasize unarmed self-defense against armed and armored foes during Feudal Japan, ie keeping the hands low down to waist area in order to immediately seize the opponent's weapon (usually worn around the waist) or grab chinks of the armor to throw them off balance then apply a lock on a limb. Kodokan Jiujitsu, which combined the effective techniques from various Japanese jiujitsu styles, has a bit of ground work although they prefer hip takedowns given the lack of pieces of armor to grab on to in the era.
BJJ, in contrast, emphasizes groundwork and excels fighting from the back (due to the guard game), since in modern combat, particularly in self-defense situations, you still have a chance to win or get back up from the ground since nobody is going to have a sword or spear at hand to kill them while in the ground.
Legitimate catch wrestlers like Lou Thesz, Karl Gotch, Billy Riley, Ed Lewis, Stanley Radwan and Stu Hart may have gone on now, so the only modern true practitioners right now are Kazushi Sakuraba (within a degree or two from Karl Gotch), Josh Barnett (same), Gene Lebell (met and trained with Ed Lewis), and Tony Cecchine (under Stanley Radwan, and personally recommended by Lou Thesz). For more information about Catch wrestling techniques, Tony Cecchine's DVD set "The Lost Art of Hooking" may not only be one of the best representation of a fraction of what the art brings, but it's actually also a guilty viewing for most advanced modern day grapplers due to how effective they really are.
Anyway, I hope my input regarding my love for western-style wrestling like catch wrestling, how it influenced other grappling styles, and how it may be just as old as boxing and survived throughout time due to its continued practice and effectiveness, and I do hope it is relevant to the current discussion (although I'm starting to think that I should have made a new thread for this -- suggestions on doing so?). I also feel the need to promote catch wrestling, as although I may be repeating myself, it's actually one of the few western martial arts that survived and continually evolved for hundreds of years all the way up to the modern day, where it's still being used or its influence is still in use in other grappling arts being utilized in both combat and mixed martial arts.
I also hope that more research and interest will be shown on catch wrestling, and how it came to be, as I haven't really found any articles or texts on its history as well as all the people involved in its development.
