Does this spirit of getting a few engineers together and taping out a chip in a few months still exist anywhere?
Edit: typo
If you could get parts made affordably and have some insight into their operation, then you might see more interest. Sort of like how affordable PCB fabrication services have led to a proliferation of customized circuit boards for all sorts of purposes, including spurious art like conference badges.
I bet if it were affordable to get a working design with a bit of studying, you'd see a lot more special-purpose chips, especially in small 3-6 pin packages.
Back to the parent question ... the main issue these days is the size of a team you need to make something serious - you essentially need a million dollars or two to pay for the people, the CAD tools, and the tape out costs ... that means it's pretty hard to just knock together something that's non-trivial