Without leet coding, named algorithms (binary search, dynamic programming, and other esoterica) or data structures (only requiring knowledge of arrays, assoc arrays, loops is enough), coding interviews still allow you to demonstrate your thinking ability.
You can construct various simple problems (that require only loops and arrays) that can easily be mapped to real world problems and see how well the programmers perform (problems: that exploit the fact that input is sorted, linear search for min/max, collect data -> loopy state transition -> output state).
A fictional scenario that you can map to a real software problem, there's 7 cities, N people (program input) in city 0. 6 buses, one goes from city 0 to city 1, other from 1 to 2, ... last from 5 to 6.
Bus has a limited capacity (6 additional program inputs). Everyone needs to be at city 6. Time starts at 0. One time tick: all buses go from city X to city (X+1), unload people at city (X+1), and then go back to city X. What's the minimum time it takes for everyone to be at city 6?
There's 7 inputs (N people and bus capacities) and there's multiple programs one can write to solve the problem. There's no math tricks, no data structures.
You can write a program whose time complexity depends on the size of the inputs O(N), or just on the number of inputs O(7), or maybe something worse or in-between.
I have no idea if there is a way for one to teach/learn how to write the simplest program as soon as possible, but some will find it in 1 minute, some in 60 minutes. The amount of time it takes might be irrelevant and the only thing you're looking for is the ability to eventually think simple enough. And the thinking bit is not the only filter. The person might struggle to use the computer (but uses it well enough for these simple problems), might have no idea about the machine that runs the program, etc.