Let us take the web-shooters seriously for a moment, purely as an engineering artefact.
The tensile requirement is brutal. Swinging a person of roughly 75 kg through an arc at urban speeds means peak line loads well beyond body weight, since centripetal acceleration at the bottom of the swing stacks on top of gravity. Spider silk is genuinely competitive with steel by weight, which is the single most plausible part of the whole design.
The real problem is volume. Cartridges the size of a wristwatch battery holding enough fluid for an afternoon of city-wide travel implies a storage density that is, let us say, generous. Either the fluid expands enormously on contact with air, or the swings are much shorter than the films suggest.
Verdict: the material science is defensible, the logistics are fiction, and the failure mode of a mid-swing cartridge empty is the most terrifying thing in the franchise.