There's no reason to use an SVM over a (C)NN nowadays though.
I'd absolutely focus on ANNs if I were an academic researcher, because that's the hot flavor of the month that's going to get your career the attention it needs to bring in funding, jobs, etc. I'd also pick it for Kaggle-type stuff, where there's effectively no real penalty for generalizing poorly. Bonus points if you consume more energy to train your model than Calgary does to stay warm in the winter.
In a business setting, though, I would only default to ANNs if it were holistically the best option for the problem domain. By "holistically" I mean, "there's more to it than chasing F1 scores at all costs." The business considerations that caused Netflix to never try productionizing the prize-winning recommendation engine, for example, are always worth thinking about. Personally, I'm disinclined to look past linear models - not even as far as kernel methods - without strong reason to believe that I'm dealing with a curve that can't be straightened with a simple feature transformation. Complexity is expensive, and needless complexity is a form of technical debt.
Huh? SVMs don't perform better than NNs on less training data.
I'm sorry, but the rest of what you said is out of date and wrong. CNNs work better than SVMs for CV tasks. There's no reason to use SVMs anymore for CV, and no one in their right mind does.