I generally use about 1cm deep of olive oil when I'm making schnitzel which may or may not be the right oil but it works for me.
I generally find that if the oil is smoking, it's too hot so I tend to get it to a temperature that is very hot, but not smoking.
Once I get it to that temperature, I don't cook too many schnitzel's at once, generally 2 or maybe 3 max to a pan depending on how big your pan is.
The final thing is that you need to make sure you beat the schnitzel out nice and thin before you crumb it so it's only about 1/2 a cm thick when you're cooking it. This way it'll only take a minute or two to cook through and won't overcook the crumb.
It sounds to me like the issue may be that you're crowding the pan.
Basically, to get everything nice and brown and crispy, you need enough space for all of the steam to escape. That picture you showed has potatoes stacked on top of each other -- that means as the bottom items cook, they're going to end up steaming the items above them.
At a diner, they have a large griddle to work with -- they can really spread things out. You're not typically that lucky in a regular kitchen, as you don't have as much space, and you have a lip on the pans that'll hold the steam in.
So, either work in smaller batches, or consider recipes that use an oven -- using sheet pans instead of a pan on the stove solves much of the problem.
One other trick is that most diners don't start from raw potatoes -- maybe with hash browns, but not for home fries, you're not going to get the nice soft interior in a reasonable amount of time unless you start with a potato that's already been baked or boiled. (If you're doing things in the oven, you might be able to, but not in a pan)
Just for reference ... I have a 14" cast iron skillet that I use for home fries ... and it's about the right size for cooking a single large potato, which might be two servings, maybe three for kids. (I tend to cook carb-heavy meals).
update : I probably should've stated this directly -- you want the chunks of potato to form a single layer in the pan, with space in between them.
Best Answer
So long as the end result is cooked, you can do it however you like.
I'm assuming frozen, ready-made, supermarket hash browns [pretty much all you ever see in the UK]
& just to be silly - you could boil them… but really, no.