Dunno who or what y'all call cowboys in Texas, but neither me nor anyone I use to work cattle ever "yeehaws" cattle when we go to move, pen or catch them. And I have won my share team ropings for sure,. Never did compete in calf roping, because injuries prevent me from running very fast.
With men that know what they are doing, on horses that know what they are doing, I can move any cattle anywhere, and rarely get out of a walk. I won't work with men or horses that I don't know...much of the time I supply the horses.
I agree with
@callmefence about the dogs, however. I won't have them around, except in a few situations where the woods are to thick to ride in. Once you get the dogs involved, though, then yeah, it usually turns into a **** show. I have the horses to take care of it when it does turn into a fence-jumping, cross-country, moto-x-on-horseback if the dogs are involved,. but I'd rather the dogs stayed on the truck 99.9999% of the time. Like
@greybeard said, often it is just one or two that are crazy , but they will get the whole herd stampeding. Best to take them out first, then the others may settle some.
In a situation like the OP described, with questionable fencing and no corral, pen, or lot, I might spend a week before I caught that herd. I'd ride the place the first day, seeing how they were gonna react to a horse, and finding a good spot to set up a corral. If they settled down, easing a horse through them, we might could pen them that day. If not, though, then if there was a way to only have water in the pen, then that's what I would do, along with some feed. And as someone else suggested, shut them away from the food and water at night. Might even use a trick I learned form an old cowboy that I started catching cows with when I was teenager. He had this big ole LH steer that a baby could lead around by the ear, and people could ride. A few times when we had to get some that were high-strung like that, he'd take Jerry over there and turn him loose with them for a few days. When we'd go back, we'd just ride up to ole Jerry,. and he'd follow along behind us, and nearly always, the cattle would fall in behind Jerry, and we'd just all walk to the pens and go in.
Dunno what it would cost up there, to get someone that knew what they were doing to catch that herd. If it were close to me....40 miles or less.... I'd charge $500 to set up the corral with my panels, and the going price to catch cows like this is $250 a head. So, that's $4250 + $500. That would include us loading them and carrying them to the closest sale barn, though. Oh,, and if they had already been trying to catch them with 4-wheelrs or dogs, or those "yeehaw cowboys" like Brute and Fence described, I'd probably not fool with them.