Valid or useful or interesting notions of "random-ness" need more than basic probability notions, since the probability of 100 "heads" in flipping a fair coin has the same probability as any other specific sequence of outcomes, but is arguably implausible as a "random" outcome.
To my mind, the "Kolmogorov-Solomonoff-Chaitin" notion of "complexity" is the apt notion. This is discussed wonderfully and at length in the first part of Li-Vitanyi's book on the subject.
A crude approximation of the idea is that a "thing" is "random" if it admits no simpler description than itself (!). Yes, of course, this depends on the language/descriptive apparatus, but has provable sense when suitably qualified.
Given that most card games refer to discernible "patterns" (things with compressible descriptions), a "random" hand would be one lacking two-of-a-kind, and so on. A "random" distribution in a deck would, in particular, have no more pattern in it than might be "expected".
The question of whether there is a notion of "too-violent-to-be-random" non-pattern-forming in a given context seems to be ambiguous: while long runs of all heads or all tails are suspicious, lack of them is also suspicious. This kind of example suggests that a configuration of a deck of cards to that no one has a playable hand might also be suspicious... depending on the context.
The operationally significant question of whether or not an innocent-seeming "mixing" can produce "randomness" with relatively few iterations is slightly different. However, from the viewpoint of "complexity", surely the answer is "no", since the hands-of-cards which arise in this way immediately admit a much simpler description than themselves. Nevertheless, or perhaps because of this observation, we can decide to declare a merely relative notion of randomness for a deck of cards, in terms of a small proper subset of "genuine" tests of "randomness/compressibility".
Of course, if the only "deals" of hands of cards that were allowed were "random" in any strong sense, the probability would be very low that anyone would have a playable hand...