Perhaps you're familiar with physicist Richard Feynman explaining why he can't explain magnetism in some other way:
"I can't explain that attraction in terms of anything else that's familiar to you. For example, if I said the magnets attract like as if they were connected by rubber bands, I would be cheating you. Because they're not connected by rubber bands ... and if you were curious enough, you'd ask me why rubber bands tend to pull back together again, and I would end up explaining that in terms of electrical forces, which are the very things that I'm trying to use the rubber bands to explain, so I have cheated very badly, you see." Richard Feynman BBC Video
The reason they are explained in binary terms is because they are actually done in binary: in the code where the host decides if two addresses are in the same network, there are bitwise boolean operators (AND, OR, NOT etc).
The operation of comparing two addresses to see if they're in the same network is typically only three machine instructions:
xor d1, d0 ; "bitwise not-equal" on two addresses
and d2, d0 ; keep network bits
beq doit ; go if same
This is the reason why address calculations are done this way: it has to be very efficient because this is done several times for every packet in every router between hosts. At the time the internet protocols were designed, it wasn't obvious that this was a good idea, in contrast to the virtual circuits of X.25 etc.
It's perfectly practical for humans to do the arithmetic in binary, hex, or 4-digit base-256 numbers but most of us aren't very good at it. Any method that isn't done in a base which is a power-of-2 is adding complexity that doesn't really exist. The fundamental idea is that the operations are fantastically simple: absolutely as simple as the engineers could make them.
The method you describe is simply a way of explaining the binary operations. I promise you people have thought about these things in every way imaginable. If you helps you understand it, then great. Otherwise I recommend understanding it directly for what it is.
The canonical answer for network addressing and masks etc is How do you calculate the prefix, network, subnet, and host numbers?
PS. Those machine instructions are approximately equivalent to C
if (!((addr1 ^ addr2) & mask)) { doit(); }