The removed instructions were binary-coded-decimal instructions, push / pop of CS/DS/ES/SS. These are minor instructions, but if your program used them, that alone means it no longer works, and that was the point.
BCD instructions are slower than emulating them with more traditional operations and were rarely used in 32-bit software (16-bit is a different story). Most people would recompile their software and never experience this as an issue.
The removal of the segment registers was the only one that really reduced capabilities - VMWare's original solution became impossible because of that. However they were replaced by hypervisor instruction sets which are much more capable. The proposed solution in this article would never have fixed this though - which again is why its true but misleading.
Let me just note that the article does not propose a solution to VMWare's problem. It only notes that when instructions are removed, code using it stops working.
Instructions were never removed from 32-bit mode though. You can run your 32-bit programs the same as before, even on today's most modern processor. You just won't get any of that 64-bit goodness.
The removal of the segment registers was the only one that really reduced capabilities - VMWare's original solution became impossible because of that. However they were replaced by hypervisor instruction sets which are much more capable. The proposed solution in this article would never have fixed this though - which again is why its true but misleading.