Not only can you go through a PhD program without taking History of Economic thought, it's hard not to. Many programs don't even offer one and aside from George Mason, I don't think it's a core part of any programs.

What I feel Austrian's tend to miss, or at least misstate, is the valuable role of abstract mathematics in economic theory. While the same tools of applied analysis that are so useful in physics are rightly criticized, the set theory and ordinal analysis used by Kenneth Arrow is extremely powerful and goes unsung.

Not coincidentally, proper use of higher mathematics tends to find results unpopular with the regime, like the impossibility of designing a fair voting system. Or, relatedly, that talking about aggregate welfare is inherently nonsense.