Having worn western wear all my adult life, I can chime in with a few points:

-the tall, cut back heel serves two purposes, one, it helps keep your foot from slipping through the typical western saddle stirrup. Second, the slanted heel makes it easier to rock back on your heels and dig the heel edges into the ground for traction when pulling on a horse or cow while on foot. They also make you walk like a cowboy (the ground shakes as you stomp across the landscape). You can also rock back onto your heels and walk through puddles like you're wearing little stilts. Can't do that with straight sided block heels.

If you're wearing sneakers on horseback, and your foot falls through the stirrup, and you slide off the horse, you're in extreme danger. The horse may bolt, and you won't be able to get your foot out of the stirrup, and you'll be dragged till the saddle girth breaks, or the horse gets tired. And you'll be severely injured or dead from being dragged a mile or so. English style saddles hold the stirrup leathers on with a backwards facing "L" hook that allows the stirrup leather to slide off if you get dragged.

- the pointed toe makes it easy to get your foot into the stirrup. Think spear tip into a funnel. A rounded or blunt toe tends to swat the stirrup away if you don't get the boot lined up perfectly with the stirrup opening.

The tall, wide, dipped top, stitched upper part of the boot does several things. First, protection for the leg, second, the stitching keeps the upper stiff so it will stand up on its own, and not sag/flop down. This, combined with the dipped top, allows your leg to move back and forth inside the upper, forcing air to pump in and out of the boot upper for cooling purposes. This is useful in the hot southwestern climate the boots were typically worn in. English riding boots don't allow this kind of air circulation. They're better suited to colder climates.

The classic cowboy boot is meant for riding, and working cattle from horseback, and some stock handling on foot. They're not meant for mud, either. Your horse handled walking through mud for you.

Modern day cowboys do most of their work in the stockyard, on foot. Classic cowboy boots are not well suited for that kind of work. Sneakers/workboots are much better. Classic cowboy boots are for old time cowboys and cubical cowboys like myself. We both do a lot of sitting.

Dress cowboy boots are a stylized version of the traditional boot. With lots of fancy stitching, designs, etc. Meant for show.

Packers are for riders that spend a lot of time walking, as in mule packers leading mule trains around, and occasionally riding in the saddle.

I would not wear Ropers in the saddle unless the saddle had breakaway stirrups.

All my riding is foxhunting, so I wear traditional English hunt clothes, with tall riding boots with low heels, and thus use an english saddle with breakaway stirrups.