As a kid I had a TI 99/4a. The TMS9900 processor didn't have any registers, it had a "workspace pointer" which let you treat a block of RAM as your registers. This was slow, but in theory allowed for convenient context switches as you'd just load a new workspace pointer.
Do any modern CPUs still use an approach like this?
Not if the CPU runs faster than 10 MHz or so. Fundamentally the speed of CPUs has gone up much, much faster than the speed of RAM for the reasons listed in the article. Some micro-controllers can still do things like you describe, but anything you'd think of as a modern CPU uses some form of caching that makes things more complicated than that.
Do any modern CPUs still use an approach like this?