I don't know of any formal Horatio Alger stories to the effect, but they basically did. Owning a person, even if it was just one, immediately increased your status in the way that home ownership might today. It meant that you were a person of substance, who mattered. The vision of a wife, 2.5 kids, and white picket fences in the suburbs current in the South at the time is basically a wife, a few sons, and at least a self-sufficient farm with a few enslaved hands. If you owned someone, you had it made.

On a personal level, owning someone would immediately transform your domestic life. Even just one means that your wife can be restricted to more traditionally feminine duties while you and the enslaved person work the farm with the kids and maybe some male relations. As more enslaved people were bought, the family would progressively transition into more leisure for the whites. A second hand might mean that the eldest son can go to school or away to university. It scales up from there and white Americans of the era are usually very sanguine about their prospects.

Socially, having an enslaved person proves that you can do it. You've got the cash, or at least the credit. More importantly, it demonstrates your manhood on multiple levels. Becoming a master makes one one's own man, not under the control of others. Instead, you're part of the civilization-defining system responsible to yourself alone. That is entailed in having the money to do it, but it goes beyond that into what kind of person you are. Nonslaveholders are most certainly white, but joining the enslaving class lets you become white and thus free on a whole new level. The informal mastery over nonwhites you had before becomes something you can perform constantly.

This is from the pen of Jefferson McKinney, who worked as an overseer for other men much of his life. He bought his first enslaved person, a woman, after saving up and expected to be in debt for a while but he had made a self-conscious investment in his future. He writes:

She is sixteen years old in May and is verry wel grone [...] If she should breed she wil be cheap in a few years and if she does not she wil always be a deer Negro besides it is getting time that I should begin to think of old age as my hed is past silvering and if eber I can get her paid for and then git a boy I intend then to quit Red River and return to St. Helena or somewhere East up the Mississippi Ribber and settle myself for life.

The spellings are in original.

Women were usually cheaper than men, so they're a budget way to get in on enslaving. Much of their value is in their expected fertility, since all their babies will be the slaves of the man who enslaved them. McKinney hoped that either using the stolen lives of her children himself or selling them to others would give him enough cash to provide for retirement and let him move back home to live near or with his brother like they did as children. Like thousands of others, he staked his future on slavery.

A J.H. Lucas wrote similarly of his success. His father fronted him enough cash for five enslaved people and some land. He cleared the land, then flipped it and two of the people at a profit.

Since then I bought at auction a boy of 17 years of age for $1100, which makes me including a girl I got by my wife six likely slaves between 14 and 22 years of age all either natives of the country or acclimated by several years stay in this country, all trained to the culture of cotton.

He's literally writing his Dad, telling him about his new stuff with an implied "aren't you proud of me?"

As Walter Johnson -I'm getting the quotes from his Soul by Soul notes, "which makes me" is a telling phrase. It does literally account for Lucas' human property, but he and the rest chose a life in the enslaving class. The people they enslaved bought them upward mobility. They were made financially. They demonstrated their savvy by picking out slaves at a good price and sorting the "likely" (who would turn them a profit at the task at hand) from the sick, from the likely rebels, the aged or otherwise infirm. That took mastering a set of skills for measuring up people against the efforts of the enslaved to influence who did or didn't buy them by presenting themselves as better or worse according to their read on potential buyers plus watching the market to know good rates.

Once you cross that Rubicon, the system feeds back on itself. If you have financial problems, buying more slaves can increase your profits. You can also rent them out if you have too many or sell them off to meet debts. If that works out for you, then success can breed success. Enslaved artisans at your plantation reduce your dependence on outside forces by giving you on-demand carpentry, blacksmiths, and so forth. When you don't need them for that job, they can be harvesting or planting. Then, of course, the operation becomes its own justification. One can have so many acres and not enough people to farm it all, so one "must" buy more.

Management of all that labor involved mastery directly, the freedom to do unto others as roughly as one liked for one's own benefit. It also invited new enslavers into mentorship with their local enslaving community and/or remotely through magazines and newspapers where they would swap tips and advice about how to best steal the labor of those they owned. Managing the enslaved well -by white standards- demonstrated further your manly, white virtue. It's a little like having a nice house or an immaculate lawn, except with effective torture of people for gain as the main mode of performance. Men would dream of well-run, profitable plantations for years as they scrimped, saved, and worked out how to make it happen.

Here's another example from Johnson, of one John Knight. He wrote a long order "of ages, sexes, and body parts" to his father-in-law who would buy the people for him in Maryland. He wanted sixty, enough to work the land without demanding too much (by his standards) and giving them "every necessary indulgence" (ditto):

("I am now having built a large number of first-rate additional Negroe quarters ...where they can keep themselves clean, comfortable, and I hope Healthy.), their working conditions ("during the heat of the day, say from 11 O'Clock A.M. to 4 O'Clock P.M. they must be in their quarters or in the shade"), health care ("there is a good physician residing ...on a removed but very convenient part of my plantation), and oversight ("I derive much satisfaction from the knowledge that I have a humane and just manager in charge of them").

Knight already owned the land, but he needed the people and dreamed big of what he would do to them. He, and many antebellum Southerners, probably didn't need Alger in fiction. They could look around themselves -even at themselves- and see successful men made of slaves. That's not to say that no such publications existed. Descriptions of newly-opened frontiers tend to stress how they can be used by white men for economic mobility. Go west, get land on the cheap, and improve it into something that can either be resold at a profit or support one in considerable comfort and independence is the white American vision for millions, whether they intend that land to be worked by someone they own or not.