OK another, slightly different take on this:
A first basic problem is the phrase "due to [random] chance". The idea of unspecified 'chance' comes naturally to students but it is hazardous for thinking clearly about uncertainty and catastrophic for doing sensible statistics. With something like a sequence of coin flips it is easy to assume that 'chance' is described by the Binomial setup with a probability of 0.5. There is a certain naturalness to it for sure, but from a statistical point of view it's not more natural than assuming 0.6 or something else. And for other less 'obvious' examples, e.g. involving real parameters it's utterly unhelpful to think about what 'chance' would look like.
With respect to the question, the key idea is understanding what sort of 'chance' is described by H0, i.e. what actual likelihood/DGP H0 names. Once that concept is in place, students finally stop talking about things happening 'by chance', and start asking what H0 actually is. (They also figure out that things can be consistent with a rather wide variety of Hs so they get a head start on confidence intervals, via inverted tests).
The second problem is that if you're on the way to Fisher's definition of p-values, you should (imho) always explain it first in terms of the data's consistency with H0 because the point of p is to see that, not to interpret the tail area as some sort of 'chance' activity, (or frankly to interpret it at all). This is purely a matter of rhetorical emphasis, obviously, but it seems to help.
In short, the harm is that this way of describing things will not generalise to any non-trivial model they might subsequently try to think about. At worst it may just add to sense of mystery that the study of statistics already generates in the sorts of people such bowdlerised descriptions are aimed at.