What's the point of the test set?
The point of a test set is to give you a final, unbiased performance measure of your entire model building process. This includes all modelling decisions in your pipeline, so any preprocessing, algorithm selection, feature engineering, feature selection, hyper parameter tuning and how you trained your model in general (5 fold? Bootstrapping? etc.). All of these decisions can lead to overfitting; for instance, selecting a set of hyperparameters that are coincidentally optimal for a particular validation set but not for the general population. If we have no test set you would not be able to identify this and would potentially be reporting highly optimistic scores.

Also, because the above modelling pipeline can get very complex, the possibility of leaking data and overfitting becomes very high. If you tune to your validation set, how will you know if your entire modelling process is not leaking data (and therefore overfitting?)

You bring up a good point; of course if we see that the test set score is poor then we will probably go back and tweak again. Thus, this just demotes the test set into a validation one if you use it too many times as you now run into the possibility of overfitting the test set (see almost every Kaggle competition). However, through repeated test set evaluation (train the model, then test it, then repeat with a different partioning) you will at least get a gauge on how variable your model is to help mitigate this problem. The amount of times you repeat will depend on how much the test set scores vary and how much uncertainty you are willing to accept (also time constraints).

In my opinion, in the business setting you should always make time to properly test your model. The dangers of overfitting are way too high and even worse; you would not even know it. If the test set scores end up being "trash" then at least you know the model is trash and you don't use it and/or you change your approach. This is way better than thinking the model is fantastic based off non rigorous validation and then having the model fail in production. The scientific method is there for a reason right?