I believe that approach comes naturally to us software developers, as we tend to think about mathematics in procedural terms (which is how programs run in the execution environment) rather than in string rewriting (which is how mathematical proofs are made). I also consider infinity as a process that never finds its ending condition, my mental model is the "infinite loop".

"ω" is not almost 0, that would be epsilon "ε". Omega ω is a number higher than any other natural number, i.e. "to the right" of the infinite line of numbers.

Mathematicians "will" this number into existence, so to say, starting from a contradiction. It's the same that they do with irrational numbers ("imagine there's this number "i" that, when multiplied by itself, it gives you -1"). With infinity, it's like: "you know this process that never ends? Well, imagine that it finishes, and let's call the result "ω".

Once they have this new number defined, they do lots of mathematical operations with it, trying to find its properties. What they never remember again after that is that the number dit not appear as the result of following the initial process to completion; they had to assume that it existed independently from the process.

* BTW, this is also why they have different kinds of infinities. They are using different never-ending processes in their respective definitions, and using the same name for all of them.