Changing the model wouldn't change anything. A turing machine is equivalent to any computational machine you want to put in there. It doesn't matter what the machine is. The turing machine is the goto because it was first used for them.
Any other machine that can simulate a turing machine can simulate failing in those same ways. You can't escape the problems of turing completeness without losing it.
I think interaction nets [0] is actually a simpler model and can simulate Turing Machine efficiently. I wish courses in Computability/Complexity theory would be taught in interaction nets instead of Turing Machines as the program written would be so much nicer and compose easily. It also has real-life uses in compiling functional languages. [1]
Symmetric interaction combinators (an instance of interaction nets like 2-state 3-symbol Turing machine is a particular type of Turing Machine) only has 3 agents and 3 rewriting rules. This makes it about as simple as lambda calculus with 3 possible terms (variable, application, and abstraction) and α-equivalence, β-reduction, and η-reduction. The advantage of interaction nets is that each rewriting rule is local and "atomic" unlike subsitution step in β-reduction which can take arbitrarily long time as it is defined recursively.
Any other machine that can simulate a turing machine can simulate failing in those same ways. You can't escape the problems of turing completeness without losing it.