Any significantly detailed description of the history of the last two centuries or so is inevitably going to make mention of some of the great social changes that have unfolded across the world. Unions and other labour movements rising throughout the 1800s and then being broken under Thatcher. Civil rights-type movements have gradually given more equality to minorities. Voting rights have expanded from being restricted to rich landowners in the early 1800s to nearly everyone today.
The High Middle Ages covered a similar length of time as the French Revolution until the present does, and humans born in 1000 wouldn't like being oppressed any more than those born in 2000 - at various points during this period there must have been efforts to 'change the rules'. Yet when the HMA are discussed, there is often very little mention of any social movements other than the Magna Carta (which is often presented as a reaction to heightening taxes and then not discussed again). Why does this perception of the HMA as unchanging exist, and what would a more realistic view of the period look like?
I'm curious where you're getting your impression that it's felt that the High Middle Ages are a monolith from. It's certainly not the accepted view in mainstream scholarship, the High Middle Ages is a time of significant change, upheaval, and excitement in Europe. It's also a time of significant artistic, literary, and intellectual development, as shown in the oft discussed "Twelfth Century Renaissance".
For my own area, the two movements that are probably most relevant to the kinds of changes you seem interested in are the Peace of God and the Truce of God. These were populist movements that attempted to limit the intermittent violence that plagued the period. The Peace of God tried to limit violence against noncombatants (commoners, women, clergy), while the Truce of God tried to prevent warfare during certain times of year (like Lent for example). These weren't universally effective, but they became popular and were influential on the development of the Crusades - a move by the church to redirect secular violence outside of Christendom (among other things).
These are hardly the only popular movements of the time. The clerical reform movements, most famously those of the Abbey of Cluny but also the development of the Imperial Church system, were an expression of greater independence for the church from secular rulers. This period also saw the development of the Franciscan and Dominican Friars , as well as the Cistercian monks, all of which took on a more ascetic lifestyle than the (at the time) dominant religious orders. On the other end, you also have the rise of the heretical Cathars/Albigensians, whose origin is controversial but should be seen at least in part as a push against the state of the church at the time.
You also have Leo IX, Gregory VII, and other reform popes redefining what the Papacy meant and creating something that we would recognise as familiar. The Capetian Kings of France rising from being practically middling nobility to actually the sort of dominant monarch that people imagine when they think of the King of France. William the Conqueror rose from bastard son, to Duke, to King of England and his descendants would expand their lands until they ruled most of France - and then just as abruptly lose almost all of it.
That's only a short list of events and movements from the period - and a gross oversimplification of them at that. The High Middle Ages was a very dynamic period of history, and I think the current state of scholarship reflects that (and has for some time), so I'm not sure where you get the impression that it wasn't.
There's some good suggested reading on the period on the AskHistorians book list which might be of interest.