Is it a problem that "successful" machine learning algorithms have large VC dimension?
When left with a discrepancy between theory and data, data is king.  Theory is intended to be predictive -- to make predictions about the world -- but when it fails to predict what we actually observe and experience, when its predictions disagree with our experience, then there is obviously something lacking in the theory.

In this case, VC theory just isn't adequate to understand modern practice in machine learning.

Unfortunately, VC theory ignores methods like regularization.  Regularization is widely used in practice, so that's a pretty important gap in VC theory.  VC theory counts the number (size, dimension) of possible models, and treats all of them as "equally valid/likely".

When we train a model with regularization, we depart from that paradigm.  Regularization implicitly encodes the assumption that "all else being equal, simpler models (explanations) are more likely to be correct".  In other words, regularization is essentially an application of Occam's Razor.  In effect, regularization encodes some kind of prior on the distribution of likely models: not all models are equally likely; simpler models are more likely to be right.  Classical VC theory doesn't take that into account, and thus can't make useful predictions about the behavior of machine learning methods that use regularization.

Practitioners aren't "brushing aside" the theory.  Rather, VC dimension simply doesn't seem to be super-relevant to practice.  It is too limited.

It's still an open question to understand why techniques like deep learning work so well.  VC dimension was an early attempt to understand machine learning -- a powerful, beautiful, valiant attempt, one that may still be of some interest -- but ultimately one that doesn't seem to give us the whole picture, perhaps partly because it does not account for things like regularization and our priors on the model.