Marked formality did make the 18th c. sometimes a rather odd world. Some extreme examples were found at the court of Louis XIV. Ultimate authority resided with the king, and so there was a constant striving in the court to advance in standing, in order to be closer to the king, and advancement was done by creating and obeying increasingly complex rules: whether to knock at a door several times or just once, and whether both doors should be opened for the visitor, or just one. Madame de Sévigné might devote a paragraph in a letter to the impertinence of someone just below her in rank who arrived at a soirée at Versailles and took a chair, forcing her to sit on a stool: or even stand! This is the strangely mannered world that Rousseau would complain of, and which even the nobles themselves would go outside and imitate shepherds and shepherdesses, in order to escape.

But outside of all that court etiquette, you do find a lot of formality in even the backwoods of America. George Washington would be set to copying polite formal letters, as a part of his schoolwork. And unschooled people would try to imitate that formality: the entrepreneur John Fitch, in his striving to create a working steamboat, tried constantly to utilize the formal rhetoric of his day to argue his cause but could never manage to keep it from sounding a little rough and ragged. And though a rich eccentric like Timothy Dexter could dump all pretense of form and manners in his writing , most people would make the attempt. So, why bother?

We can’t really explain human behavior only in terms of functionality, and I wouldn’t dare to try to tie 18th c. manners solely to commerce. But a lot of this, both for the French court and for backwoods America, was a matter of signifying trustworthiness. And trust was a vitally important part of getting along in their world. This was not a transparent economy, with credit cards and bank balances, credit scores, and being able to Google someone to see if they’re a fugitive from justice. It was difficult to research people immediately. It was important to look for signifiers. Because you needed to trust them.

Are you a merchant, operating something like a general store in rural Virginia circa 1790? Or a blacksmith? Most all your customers are going to be buying on credit. They will come and make purchases, take goods away: and those will be entered into their accounts in your ledger. They may also bring in items to put on that account- maybe one will have sawn some planks, from logs felled on his woodlot. Maybe another will have distilled a hundred gallons of whiskey. Those are entered as credits, against their debts. Then , the customers would periodically settle up. Usually when the harvest was in, and sold, because most of them would be farmers. So, much of the time, a transaction would not be completed right then and there- you had to initially assess them, in order to trust that they would regularly pay on their accounts. And if they presented themselves well, you would be more likely to initially approve them. Now, of course, if a stranger walked into a tavern and ordered a beer, he would be asked to pay in cash (and if he arrived at the tavern on foot, instead of riding up, he’d likely be asked to pay up front before he got his beer). But the locals would be keeping accounts.

And they would be “passing notes”. Not letters, but IOU’s. A woodsman drops off a load of firewood for you. You write him a note, saying he’s to get 25 cents. He presents that to the farmer he owes, who’s got the woodlot he uses. That farmer brings the note back to your store, and it goes onto his account as a credit. And a note could go far. In the cash-poor 1780's, farmers and merchants in western Massachusetts were passing notes and keeping accounts so much that they revolted when forced to pay their taxes in cash ( Shay's Rebellion)

And your suppliers- perhaps in Philadelphia, or Baltimore- might extend credit to you as well. And how would they know? You would have gotten to know the local customers in your area, who you could give credit, what they were like. What about a wholesaler in Baltimore? Perhaps this is a very important man. Do you come up to Mr. Big on the street and say hello? Of course not. Asking him to assess you on the spot would be difficult for him. And, if he refused to return your greeting, even pretended you were not there, the public snub might not only affect your ability to ever get credit from him, but might also be noticed by other merchants- and you might have trouble getting yourself credit with them as well. No: you find someone to make an introduction, someone whom he knows, who can tell Mr Big that you would like to see him: and of course you will present yourself properly, when you meet.. Perhaps that’s simply a matter of asking Mr Big’s secretary for an appointment, but even then, that secretary is going to assess you. Far better to find some mutual friend, who can vouch for you.

What if you have a big project, like an iron mine? You need investors. Three are found, ready to put in , say, $1,000 each in return for a quarter of the profits. One of them is Mr Big. They put in their money, the pit is started, tools are bought, workers are hired and iron ore is just starting to get piled up when it turns out that Mr Big had too many debts, one of his creditors has gone before a judge and the sheriff is at his house to take all the household goods, and the other creditors have panicked and done the same, looking for his assets. They appear at your mine, and start hauling away anything that might amount to $1,000. Your mine, so to speak, is sunk. How could you have known? You would hope to have your friends and other business contacts tell you- people you can trust.

Again, I don't want to draw the simple conclusion that the advent of modern communications made it possible for easier actual verification to replace signifiers of trust, so it's now possible for a bank president on Informal Friday to be seen wearing her jeans at an office meeting. But those formal manners of the period did serve a purpose. They signaled that someone was trying to obey societal rules, behave properly, and could be trusted.

Although focused most specifically on the duel, this older answer of mine may be of interest as the rising doctrine of civility was an integral part of the phenomenon, so covered a good deal.