How do I justify the cost savings of my recommendations?

You can find the right fix now. The harder part is showing it pays for itself, and that is a different skill than spotting the problem. Speaking in numbers is not a betrayal of the craft. It is translating what you already see into the language the person with the budget actually hears. Decision-makers don't fund good ideas; they fund good returns. So put your recommendation in their terms. What does the problem cost them each month it goes unfixed? What does the fix cost, and how fast does it pay back? Even rough numbers beat none, because they turn a judgment call into a comparison anyone can follow. Take your next recommendation and write one line: this costs X, it saves Y per year, it pays for itself in Z. If you don't know the numbers, that is your first task, not an afterthought. Once you can show the payback plainly, you will find the yes comes far more often.