I think Burroughs mainframes also used stack architectures. I suspect the architecture wasn't entirely responsible for Burroughs' demise, but they seem to be an outlier.

I imagine one potential problem with stack architectures is running out of stack memory and having to spill to main memory. (Presumably using main memory for the stack would be too slow.) This seems unlikely for a single thread but I could imagine it being a big issue with thousands of threads. Context switches where you have to save the whole stack to memory are going to be more like cache misses or page faults.

Another potential issue might be dealing with data that isn't on the top of the stack. A solution could be to add a deep stack addressing mode but then you're basically mixing in a register machine and the simplicity and orthogonality might be lost.

The interesting thing to me is that several stack-oriented interpreters, compilers, and runtimes (including Forth, Pascal, Smalltalk, Java, and WebAssembly systems) have been developed, so maybe a stack-based CPU might be a good match for those systems.