Many years ago I sat next to the CTO of Broadcom on a plane

This was probably not true many years ago, or at least was a simplification. Even with some automated routing, there are many subtleties like high frequency electronics; not trying to have your radio equipment routing through your power supply, etc.

They even have their own languages for coding the chip logic ( Verilog is one). My partner did chip design and layout. The used a tool call magma (We have some mugs from a conference she went to) but they got bought by synopsis. Very expensive, very specialized software https://www.synopsys.com/#

It is a mix: digital logic is almost entirely laid out by machine, but analog circuits and the underlying logic cells (as well as certain specialised digital blocks like memories) are a lot less automated. The most expensive part isn't doing that: it's doing the validation and simulation of the design so that it'll have the best chance of working after you've spend 5-6 figures and many months on actually building it (and then there's the cost of testing it in reality and figuring out what isn't actually working and how to fix it: the first batch of chips still extremely rarely works).