What you said certainly works, but I'm not sure computability is actually the biggest issue here?

Have a look at how SAT solvers or Mixed Integer Linear Programming solvers are used.

There you specify a clear goal (with your code), and then you let the solvers run. You can, but you don't need to, let the solvers run all the way to optimality. And the solvers are also allowed to use all kinds of heuristics to find their answers, but that doesn't impact the statement of your objective.

Compare that to how many people write code without solvers: the objective of what your code is trying to achieve is seldom clearly spelled out, and is instead mixed up with the how-to-compute bits, including all the compromises and heuristics you make to get a reasonable runtime or to accommodate some changes in the spec your boss asked for at the last minute.

Using a solver ain't formal verification, but it shows the same separation between spec and implementation.

Another benefit of formal verification, that you already imply: your formal verification doesn't have to determine the behaviour of your software, and you can have multiple specs simultaneously. But you can only have a single implementation active at a time (even if you use a high level implementation language.)

So you can add 'handling a user request must terminate in finite time' as a (partial) spec. It's an important property, but it tells you almost nothing about the required business logic. In addition you can add "users shouldn't be able to withdraw more than they deposited" (and other more complicated rules), and you only have to review these rules once, and don't have to touch them again, even when you implement a clever new money transfer routine.