It's likely that they've seen deep learning used to solve problems in the past that are more suitable to a simple regression. They may not have seen it used to solve problems where regressions failed.
In your place, I'd dive into why they feel the way they do. Maybe they're right! More likely, maybe you're using deep learning for jobs it isn't best suited.
This attitude typically isn't about the "best tool for the job", i.e. do I really need DL or are decision trees or some regression sufficient for what I need to achieve? But about the persistent "you don't need DL at all!" stance.
In a way I understand it, for example if somebody finished their computer vision PhD before Deep Learning and don't want to admit their knowledge is now next to useless for most industry cases...
It's likely that they've seen deep learning used to solve problems in the past that are more suitable to a simple regression. They may not have seen it used to solve problems where regressions failed.
In your place, I'd dive into why they feel the way they do. Maybe they're right! More likely, maybe you're using deep learning for jobs it isn't best suited.