The invention of computing automata and the techniques required to program them, such as defining semantics, settles this issue concretely in favor of Hilbert.
The state space is far too large to show a program is correct by constructing all possible processes it could execute. Instead you must show that the program admits no incorrect processes whatsoever, and as far as I know mathematical formalism is the only possible way to do that.
Of course if one just wants to make money, formal correctness is observably of absolutely no importance. And there's nothing wrong with wanting to make money!
It always seems to me that using the LEM over infinite sets is akin to claiming that all problems are decidable.