Re: What we would actually need...
Well actually there are some points to that.
First of all, yes you can modify a processor, however that's going to be _really_ hard as you are at an extremely low level. You are at a gate level and try to find out where, in unknown future revisions of the software, you have to do something in order to achieve your goal... while still conforming to the published specs which are very tight. The examples shown in academic paper assume known code which makes it far easier.
So if I was a government I'd go the route via some sort of "security enclave", essentially a separate system hidden from the rest of the system that can run software that patches future unknown code. That's far more realistic to pull off.
BTW you can actually buy processors which are so tightly speced and so simple in construction that you can make reasonably sure there were no malevolent actors involved. The 6502, the Z80 or the ATMega microcontrollers are prime examples for this. So if you can live with a small 8-Bit system, you can be reasonably sure it'll be safe.