My overall advice is this:
Your recipe is solid!
Just think of each step as its own dish you'll combine at the end, instead of going for a one-shot one-pot.
Then just get one high-sided pan like this:
for your saucy skillet dishes.
I am a believer in brining.
Salt draws moisture out of food, which is what you want: a drier potato.
Likewise, it's what you don't want, at the end of cooking: salt on the outside of a crispy potato pulls the moisture out and it can reside on the fry surface, making it soggy instead of crispy.
That's why, if you want to make sure you always get fresh fries at McDonald's, ask for your order to have no salt, then add salt yourself right before you eat them.
In terms of how I do fries at home without even brining, I take peanut oil, fill my small wok about 75% of the way up, turn the heat on high until it hits 350F, and use cheap pre-cut frozen potato.
When they go in the oil, the temperature will drop to 275F or 280F: then you just let them deep fry them until they're whistling (back up around 350F)and you're all good.
A spider basket helps to fish them out.
They're restaurant-quality (and nothing makes you need restaurants less, like being able to make your own fries at home).