A typical website is nowhere near as complex as a human body though.

Isn't it more that case that certain websites have specific requirements that are beyond and average web developer - for example if the have massive traffic, or stream huge amounts of data, or implement a complex SPA, or have stringent security requirements? Sites like that might require specialist engineers to handle those requirements.

A run-of-the-mill shopping website or similar, with moderate traffic (ie the vast majority of sites) can still be run by a small team who each understand the full stack.