I do think that people that think that ML = big data are mistaken. ML is about making the most of your data, however much of it you have.
In my view, the main difference between ML and plain statistics is that with the latter, you come up with the appropriate model apriori, and then make sure the data satisfies the assumptions so that you can draw conclusions from the in-sample fit of the model. You control for overfitting by choosing the simplest model that is reasonable - often univariate linear regression.
Whereas with ML, you let the data dictate how complex a model you should use. You choose the appropriate model complexity using techniques like cross-validation, and verify the effectiveness of your model empirically.
ML is often used interchangeably with ANNs which I think is a mistake. Take structured data problems on Kaggle and you would very rarely see ANNs as a major predictor in the winning models.
So if you've got an analyst sitting on that issue - then your problem is solved anyway.
So again, why hire somebody with a trendy specialty who is <probably> full of BS, when you can hire someone with some respect for classics?