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.
Americas test kitchen has tested this in a prior issue. I cannot put my hands on the issue right now which explained their process and the results. It also contained a good deal of explanation about the science of the process. They tried all sorts of liquids in their omelette and scrambled eggs recipe. They found that water did make the eggs slightly fluffy but did not slow the coagulation process. This led to a tougher scramble than milk. They wanted the same result for an omelette with a denser texture. The end result being that butter was better for an omelet.
However, some of the result was later reported in a Smithsonian Magazine Article. It is a much more compact summary but fairly on the nose for your question.
Their answer was as follows:
Add milk to scrambled eggs, frozen butter to omelets: If you want
scrambled eggs, most of us know to throw in a bit of milk or butter
while scrambling. That’s because the lipids in the dairy coat the
proteins in the egg (11 percent in the whites and 16 percent in the
yolks) and slow down the process of coagulation, a.k.a. when the
proteins are denatured and unfurl, releasing much of the water in the
mixture. Adding fat helps keep some moisture in and fluff up the final
product. But the same does not go for omelets. “While scrambled eggs
should be fluffy, an omelet is more compact,” the authors write.
While milk works for scrambled eggs, it can add to much moisture to an
omelet. The chefs recommend frozen bits of butter instead, which melt
more slowly and disperse more evenly. And it turns out you can go
ahead and salt the eggs before you even cook them up. Because salt
affects the electrical charge on the proteins, it weakens the bonds
between them, preventing overcoagulation. Bring that up at your next
brunch.
The Science of Good Cooking: Tips From America’s Test Kitchen
Best Answer
There is nothing wrong with this approach. In fact, it is a sensible way to ensure the eggs are cooked to your liking...just add them at the end. The only disadvantage is that you have another cooking vessel to clean.