Theoretically infinite memory isn't really the problem with reasoning about Turing-complete programs. In practice, the inability to guarantee that any program will halt still applies to any system with enough memory to do anything more than serve as an interesting toy.
I mean, I think this should be self-evident: our computers already do have finite memory. Giving a program slightly less memory to work with doesn't really change anything; you're still probably giving that statically-allocated program more memory than entire machines had in the 80s, and it's not like the limitations of computers in the 80s made us any better at reasoning about programs in general.