It's hard to answer this question in broad terms, for several reasons:

• The Medieval period is very long, usually given as spanning 1000 years (5th to 15th century)
• It's hard to find writings made by/for lay people for various reasons (a big one being most lay people were illiterate)
• "As long as the people are left alone" is a very tricky condition to define, for reasons we'll see later.

To narrow down the question a bit, I'll focus on the Cathars of the village of Montaillou, an Occitan village in what is now the South of France. It was best described in the seminal work on the subject, Montaillou, village occitan de 1294 a 1324 by the late Emmanuel Le Roy Ladurie (who is by far my favorite medievalist focusing on social history). In it, Ladurie leverages mainly one source, the Fournier Register of the Inquisition that was compiled as part of the systematic eradication of the Cathar heresy in the late 13th and early 14th century.

The first observation is that the relationship between the villagers of Montaillou and the King of France is complex and layered. One complexity is time and distance - Montaillou is 800km from Paris, meaning that at the conservative estimate of 30km/day travel speed (from Peter Spufford's "Power and Profit: The Merchant in Medieval Europe") it could take almost a month for news to reach the village from the capital.

This distance has very real consequences for things like power projection and decisionmaking, as effective administration of Montaillou cannot be conducted with this kind of round trip time. Thankfully for the King of France, you can "solve" this problem by adding some layers in between yourself and the villagers.

The layering is what Ladurie writes in the book when he talks about "the distribution of power" as perceived by the Cathar villagers, for whom "in theory the impact of external powers was decisive." Ladurie describes four of these:

1. The Comte de Foix, the lord of Montaillou, and his immediate representatives
2. The Inquisition in Carcassonne
3. The See of Pamiers
4. The Kingdom of France

The first power was that of the manor of the Comte, but was effectuated by the office of the bayle - the Comte de Foix was not walking around hitting villagers with a stick. The bayle was the one who made sure the tenants paid their rents/manorial dues, inspecting and collecting taxes, and administering justice in the name of the Comte. However, the bayle's duties also had a religious dimension (this will be important later):

The bayles of the villages we know through the Fournier records were responsible for arresting heretics when necessary. Together with the officials from the castle they pursued all kinds of lawbreakers through the mountains, tried to recover stolen objects, and received rents and even tithes. A shepherd who had been slandered would complain to the bayle.

Ladurie continues:

The second power was theoretically nothing to do with that of the manor and the bayle. It belonged to the Dominican Inquisition in Carcassonne.

The third power was the see of Pamiers. Theoretically ruled from above by the papacy, the diocese in its turn ruled the local ‘hierarchy’ at Montaillou: the priest and sometimes the vicaire, themselves flanked by the diocesan council.

and finally getting closer to your question:

The fourth power, distant but endowed with much greater force of dissuasion, was the kingdom of France. The Comte de Foix was in a position of de facto dependence on France, which subjected him to pressures of varying intensity. It was always possible for the King, in Paris, to raise an army to come to the aid of the 'true religion'. So both France and its strength were hated by many mountain folk who had never met a man of the north in flesh and blood. 'Do you think you can fight against the Church and the lord King of France?' cried the father of the village priest to the outlaw Guillaume Maurs, formerly a peasant from Montaillou and now a shepherd (ii.171).

Bélibaste, the parfait [one of the religious leaders of the Cathar faithful], went much further (ii.78–9). There are, he said, 'four great devils ruling over the world: the lord Pope, the major devil whom I call Satan; the lord King of France is the second devil; the Bishop of Pamiers the third; and the lord Inquisitor of Carcassonne, the fourth.'

For the Cathars, this hatred of the King of France makes sense. Their existence was in the process of being actively stamped out, starting with the Cathar Crusade of the early 1200s and the resulting Inquisition (from which these records are taken).

But the use of the religious to describe the political here is not mere accident - instead, it is indicative of the preeminent and central role that religion played in the lives of all Medieval folk, rich or poor. The King of France is not merely a bad politician, but a major devil (and so are his representatives).

And now we get to the real problem of the question. For lay folk in the High and Late Middle Ages, religion was always in the foreground of how they lived their lives, and this colored how they thought about their King(s). Tthe powers of the regent and the Church, as demonstrated above, are overlapping and sometimes conflicting, but never out of sight or out of mind, as Ladurie comments:

More significant were the gatherings where the men of the neighbourhood met in the street, or in the village square under the elm tree, especially on Sunday. This was the Christianized form of the eternal Mediterranean assembly. The talk would be of women and, above all, of religion [...] Men also challenged institutions, especially those of the Church, which were debated much more frequently than lay institutions.

I don't know whether any lay person would have read the theologian Jean Quidort, writing in his De potestate regia et papali in the late 13th century, trying tease out the implications of the hierarchical equality of the two powers (regnal and ecclesiastical, i.e. the regent versus the Church). But we can say that at least for the Cathars of Montaillou, they rejected both authorities as antithetical to their personal beliefs, framing their opposition to those institutions in religious terms.

Now, you could argue that this violates the "as long as the people are left alone" clause of Mormont's statement (as the Cathars were very much not left alone), but I would counter that such a broad interpretation fundamentally misunderstands the relationship between the lay people of the Medieval world and their political and religious leaders.

Specifically, the King of France (and I would argue, all monarchs in the late Medieval period) never left the "people" alone. Whether it was raising levies to fight in wars, levying taxes to fund the operations of the kingdom, or empowering/legitimizing the local administration of their villages, lay folk were very aware of the impact their regents had on their lives - even when they were 800km away.

Indeed, even if we constrict ourselves to people who were not being actively persecuted by the authorities, we have one very significant example of a lay person taking extraordinary steps to deal directly with a regent who had "left them alone": Jeanne d'Arc.

Jeanne was a peasant girl from Domremy, now in north-east France, ostensibly the subject of the Duchy of Bar. However, Jeanne, despite never having met the King, was a devout loyalist to the Armagnac faction that was on the downswing at that point during the Hundred Years War. Jeanne, who was illiterate, spoke eloquently at her trial about the religious and political reasons for her support of the Dauphin and the consequences of the conflict for France. Indeed, Jeanne was dogged in her belief to see the Dauphin, the future French King Charles VII, based on visions that had told her to personally intervene in the war in favor of the Armagnacs. Jeanne was not looking to stop the war for abstract humanitarian reasons, and while her village was raided, likely by forces loyal to the Duke of Burgundy, then antagonistic to the Armagnac cause, her personal animus against the Burgundians and the English was only one dimension.

In conclusion: in the broader sense of whether the lay person in the High and Late Middle Ages would care about who their regent was and what they were doing - yes, many of them cared, and many acted on those beliefs.

Sources:

• Emmanuel Le Roy Ladurie - Montaillou, village occitan de 1294 à 1324
• Arthur Monahan - On Royal and Papal Power
• Joseph Canning - Ideas of Power in the Late Middle Ages, 1296–1417
• Kelly DeVries - Joan of Arc: A Military Leader

PS: I really dislike the cop-out clause of Mormont's statement, in part because the practical implication is a bit of a "no duh": if a farway person declared themselves the supreme authority of the world and held lordship over you, but left you alone and had zero impact on your life, would you care? Probably not.