The only (relatively minor) downsides I can see are:

(A) it makes the code "singleton" by default: you can't instantiate two objects of the same class with two different values for the global argument

(B) unless the global argument is declared in the same class, it slightly obfuscates the fact that a class depends on the argument (compared to all dependencies being constructor parameters)

(C) it makes testing, especially multithreaded testing, more awkward and error-prone: tests have to set and reset global variables

It's easy enough to refactor (A) where needed, as long as you have control of the code, and (C) can be worked around with good enough test helpers.

I think one can reasonably argue both for and against the convenience outweighing these downsides.