>That's left me wondering: will there be a point at which I struggle, after having spent decades writing code that powers the web?

I'd say it's not that tech will become too difficult to use, but the constant and frequent changes will start to annoy you to a point where you just don't invest as much time into learning how the new stuff works. YMMV, but I am less than half your dad's age and I'm already starting to feel it.