No, no, and no! Sword dances originated in the fourteenth century, the hare as an Eastertide symbol in the sixteenth, and "John Barleycorn" in the late fifteenth century at the earliest - and, in those early versions, he doesn't undergo the resurrection that inspires Frazer to connect the alleged folk practice with his dubious category of "life-death-rebirth deities." The thumbs-up connection is actually reasonable, except that the modern gestures come to us from later paintings which seem to have misinterpreted the written descriptions of the ancient versions, and of course we no longer use either in a way that literally calls for somebody to die violently. In truth, the supposition that modern folk customs so often preserve accurate memories of ancient practices otherwise forgotten is pure Victorian fantasy.
The Easter Bunny example is particularly notable for the fact that our only source for the existence of an Anglo-Saxon goddess named "Eostre" (*ostara is Jakob Grimm's erroneous reconstruction of the proto-Germanic form; modern scholars give *austron instead) is the Venerable Bede, who says that she is who the Anglo-Saxon heathens made sacrifices to during the month they called "Eosturmonað." The modern scholarly consensus is that Bede, a Christian living at least a generation after the near-total Christianization of England, misidentified the name of a springtime festival with the goddess it honored (who was probably Freo, corresponding in the Norse pantheon to Freya) - and, of course, Bede doesn't mention the rabbits or the eggs, as he lived nearly a thousand years before those were introduced to the holiday. The Easter hare was an invention of later German Christians as a way to celebrate the changing season, which felt to them a natural part of their most important Christian holiday.
It's certainly true that folkways can persist for a startlingly long time, as in the case of the Abbots Bromley horn dance, which dates at least back to the seventeenth century and possibly to the end of the tenth (with the thirteenth century being perhaps the most probable guess). But they do not accurately retain ancient religious significance; the horn dance is known, in the past four hundred years, to have innovated a great many features not present even then, and while the earliest version likely would have borne enough similarity to the modern that we could recognize the relationship instantly, it also is also known for certain to have lacked many distinctive features of the modern dance, and could easily have had other aspects in their place which today we would not know to expect. If, as has been speculated, it originates as a charm to improve the hunt (which would not necessarily require it to have originally been danced by non-Christians!), it has certainly lost the sympathetic magic aspect that would imply.
It's not even remotely unheard of for a folk custom to last for a thousand years, in a form that can be recognized and traced to its origin. (It may very well be true, for instance, that the modern wedding ring relates directly to the same oath-swearing customs found in the Icelandic Sagas, and the handshake really is a noticeably-modified version of a greeting used by the ancient Romans.) But it is rare enough to be remarkable each time we find an example, and and even a century works noticeable changes on the specifics.
It's far more common for people to suppose a far more ancient origin for a tradition than the evidence bears out, and to assume that it has always been done in the manner they encountered it as children with all the particulars that implies; this often leads to the presumption that specific recently-innovated details must have significance in a cultural context far earlier than the custom's origins. This is always a mistake, but it is one that since the nineteenth century and sometimes before has bedeviled even quality academic sources on the origins and early significance of modern folkways. Modern researchers are a lot better at avoiding this sort of problem, but it's still easy (especially for us lay people) to be led astray by the earlier commentators.
Another part of the problem is antiquarians of the late nineteenth century, who supposed that folk customs in rural places changed very slowly and thus would accurately reflect something old, and then posited what their Pagan significance must have been. (The Golden Bough is a masterwork of such an edifice of speculation.) Even as they recorded the changes that were underway in their own lifetimes, these early folklorists refused to consider that such change had never been absent, and so it seemed self-evident to them that customs regarded as old by the people who did them were older not only than any individual participant, but even than the very nations and towns in which they were carried out, older than the languages of the verses frequently involved or than the invention of the instruments played to accompany them. The mistake seems laughable in hindsight, but only because today we have learned not to approach serious study with the blinders of the past.
Naturally, it's not just claimed Pagan survivals in European culture which are so often inaccurate. Neither are claims that Native American practices are of any greater antiquity; when the evidence is even available for the historical development of Native folkways, it almost always points to a fascinating history rich with change and growth (just as in cultures where written records are more common) rather than the stasis sometimes condescendingly attributed to them. Chinese texts are also especially prone to making false assertions of antiquity, for at times it has been usual in Chinese culture to claim an ancient origin for a practice as a way of conveying respect for it rather than as a concrete historical assertion; modern historians working from old sources which say something is even older have often made the mistake of taking these claims at face value. Evidence for less recent transformations of the indigenous peoples of Oceania is often harder to find (with the remarkable exception of the case of rongo rongo, a writing system from Easter Island that seems to have appeared and disappeared within a window of about two centuries), but modern research rather predictably shows us cultural transformations underway no less rapidly among these cultures than throughout the rest of the world, and we should be shocked if that fact were any different four hundred or four thousand years ago.
And, of course, when a non-specialist has a ready claim of something being a survival of ancient practice that supports a point of modern political or religious interest, the evidence is almost never on their side - consider the notion that the Bible has always taught that marriage was between exactly two people, when polygamous Biblical patriarchs are so readily apparent! The point the person is making may itself be valid in a modern context, and the connection with and reification of an imagined ancient origin may very well be a resonant and relevant part of the custom in its present form, but the first reaction to any claim of pre-Christian or otherwise startlingly ancient origin for anything habitually done in the world today should always be doubt, until a primary source can be identified and double-checked.
None of this, of course, should dissuade modern neo-Pagans from accepting folkways as authentic parts of their own religion, any more than Christians have been afraid to incorporate traditions of secular origin into Christian practice over the past two millennia. But it is fair to say that the mainstream Anglo-American culture in which I was raised does not contain more than a few small customs of Pagan origin, and those have been so heavily modified over the centuries that we cannot be said to be continuing the original Pagan practice in the present day. This does not stop people from making claims about how Trick-or-Treating relates to early Celtic rites (which it doesn't; though there are distant Pagan connections with the origin of the holiday, that particular custom was invented within living memory, and has explicitly Christian antecedents) and how Christmas trees derive from Saturnalia trees (which weren't even a thing; while sacred trees are indeed a core part of Germanic paganism, trees cut down and re-erected for specific holidays are first documented in the fifteenth century, and the custom of decorating them with lights may have been invented by Martin Luther - who was decidedly not a Pagan), but the fact that such assertions prove spurious on serious investigation should surprise nobody - at least, nobody with even a passing awareness of the ever-changing nature of folkloric culture. Simply put, modern folkways do not accurately recall ancient ritual, ever.