You say they had horses for those types of traveling, except they most likely didn’t. Only a successful farmer could afford a horse. And even if they had a horse, chances were it was busy or tired for farming works to be used for transportation.
Most people walked to were they needed to go, in if the modern bicycle would have existed a farmers dream would be to have a successful crop one year so they could afford a bicycle. In fact a modern bicycle would have been a good investment for many farm as they allow farmers to bring more products to a market quicker and more efficiently, in turn allowing the transporter to be back at their farming job quicker and less tired.
This can be seen in today’s poorer regions where no trains nor road runs. But the occasional farm has invested in a bicycle to help with transportation.
We can't make a universal rule like this. Everyone has a particular mental image of how farming worked long ago, but circumstances varied. Horses can eat a wide variety of plants, including something that is growing wild right outside your door. The density of farmers probably had more to do with whether they could afford a horse than anything else had. In locations that hadn't been farmed relentlessly for centuries, there was spare pasture for a horse or two to feed themselves. For an example, just look at traditional herders. Those people are poor as dirt even today, yet they all have horses. Or look at swiddeners. They didn't move every year, but they moved often enough that not every square meter of land had been converted to intensive agriculture. Many of them, both today (when swiddening is rare) and millennia ago (much more common!) had all sorts of livestock including horses.
Many ancient farmers couldn't have owned horses because they didn't have a way to feed them. Many other ancient farmers did own horses and could feed them.
No... Farmers didn’t bring their products to market unless they were running a plantation, ie they were very large operators, which were few and far between and would certainly nit having been riding a bicycle - those guys would be going in a nice two horse cart. The small guys sold to a local buyer who handled the shipping. Part of the reason for this is that the products you produced for sale were very bulky. A tobacco bale could be over half a ton; similar for cotton. Bulk grains and beans weren’t as likely to be sold to distant buyers, but they too are too bulky to carry by bicycle.
I round up animals on a daily basis. A bicycle is worse than useless for this purpose. Which is why farmers don’t use them for rounding up animals today.
Shopping was done at most a few times a year. You rode your horse (or mule or donkey - see below) to the general store. Often, salesman came to you, specifically because you weren’t going to town much. So you’d have traveling brush salesmen - or pots and pans salesmen, traveling patent medicine salesmen, tinkers and ferriers.
When you went to town, the kinds of things you were buying were usually too bulky to carry by bike. You didn’t go to town to buy a shirt or two. You went to buy a plow, or if you had a good year some china. Popping in to town on a nike would have been a very niche use case.
Also, you weren’t buying very much equipment. What you couldn’t make yourself you had the local blacksmith make for you. There’s very little on an 1850s American farm that wasn’t locally produced.
“Only a successful farmer could afford a horse.”
Err... no... Not in the US anyway. In order to farm you had to plow and in order to plow you had to have a draft animal. That wasn’t always a horse, in some cases it was a mule or donkey, or sometimes in the west an ox. And all of those animals were also used for transport. Granted nobody rode oxen but they often pulled wagons, hence the term “oxcart”.
There was a term used coined after the Civil War: “40 acres and a mule”. It comes from field orders issued by Union General William “Tecumseh” Sherman with the approval of Abraham Lincoln, and it redistributed white property to newly freed slaves. The idea was that the slaves needed at minimum those two things to be self supporting.
But in any case, you’ve got the order wrong. You didn’t buy a horse (or a mule) once you became a successful farmer. You bought a horse in order to become a successful farmer, in much the same way that the successful programmer today will own a (or will have an employer-provided) computer. It was an essential piece of equipment.
In the latter case bringing a freshly slaughtered goat, a couple of basket worth of grain, or textiles, to town three days a week, and sell in a public market is not uncommon. In the former case it is unthinkable.
In the case of rounding up animals I can see bicycle being quite useful if you have a small stock that you grace in large swaths of shared uninhabited lands. At the start of the gracing season you might not use the bicycle, but during the gracing season you might need to move them from one area to another, then it is definitely handy to be able to bike to the first area, herd them to the next one, walk back, and bike home.
Note that neither of my examples were ever common in post colonial America, but quite common throughout Europe and are still common in sub Saharan Africa.