Yes, "West Asia" is definitely the worse term by far, as most of "West Asia":

is actually
east of the accurate definition of Asia (shown in red):

making the name absurd to anyone who knows history. "West Asia" is also functionally worse because it excludes Egypt etc., unlike "Middle East":

On the other hand, Mashriq:

has the disadvantage of excluding Iran and Turkey. The best we can do is Rashidun:

I suppose we could try to promote the term "Greater Rashidun"?