it is true that rigour is eventually needed, but the principia, laplace's celestial mechanics and countless other works (heard of euler?) were all published before cauchy and weirstrass. all of the wonderful work in elliptic functions by gauss, abel and jacobi was done before rigour was en vogue. euclidean and non-euclidean geometries both flourished wonderfully w/o modern rigour (when was hilbert's book on geometry published?)
you're also wrong about your examples. it wasn't nowhere differentiable functions, but fourier series that motivated lebesgue. that's what the author is referring to regarding analytical traps ie, monotone convergence.
it is also quite a leap to assert the arithmetical definition of limits solves the zeno paradox!!! i few of my
colleague's might disagree with you.
don't led your initial fascination with rigour (it can be addicting) get in the way of your intuition. rigour is necessary, but it comes after - sometimes to the chagrin of some. look at the teaching of modern algebra. pure abstraction and rigour, with complete detachment of all the wonderful ideas, and experiences that gave it rise.