Even if your area of study is normed spaces, it is not enough to know just about the topology derived from the norm. Weak and weak* topologies play an important role too. And different topologies are important in PDEs.

If I am teaching the concept of a topology and trying to defend it against an imaginary critic who says "All interesting topologies are metrizable" then I'll emphasize the notion of a product topology. Actually, one of the nicest examples, the product of countably many copies of {0,1}, is metrizable, but it's easy to see (i) that there is no single metric that is obviously best and (ii) that it is somehow nicer to argue directly from the topology.

PS I found algebraic topology very hard when I first met it. I think it was for exactly the kinds of reasons you describe.