I'm awfully sorry to be asking a question such as this, especially given the amount of times Schröder-Bernstein has been a highly useful citation when proving bijections or other similar things.
Nonetheless, I hope it is sufficient excuse to say that I am unfortunately a disaffected student merely trying to make sense of the purpose of what is taught to them.
Why don't we just take it as obvious that there is a bijeciton between two sets if they have at least as many elements as one another? Even setting aside antisymmetry as one of the properties which defines the $\leq$ relation, an injeciton means that you have mapped every element of a set into another. So, if every single element in one set can uniquely map to another, and the same goes for the latter, surely it's trivial that there would be a bijection between the two? Why do we need to develop a 'convoluted' proof to show this? What am I missing?