So far, scientists have been able to test spatially separated entanglement between 2 parties, and recently even between 3 parties. The latter experiment (link)(without paywall) states the following challenges:
No attempts have been made to close locality loopholes in Bell
experiments involving three or more particles. The primary reason
for this is source brightness. While entangled photon pairs have been
generated and detected at rates in excess of 1 MHz,
entangled photon triplets have only been observed at rates on the
order of, necessitating long measurement times. In
addition, further experimental challenges include high sensitivity to
loss, causality relations requiring a complex experimental set-up and
demanding stability requirements.
However, these type of experiments are interested in measuring the violation of classical bounds (e.g., proving that no classical hidden variable theories are possible, thus confirming Quantum Mechanics), such as Bell's or Mermin's inequalities. Your question is why games have not yet been experimentally demonstrated, but it turns out that testing inequalities is exactly the same.
Edit: I realized only later that quantum games and Bell-type inequalities are actually the exact same. Let me try to explain the equivalence below, and then give an example of what an experiment would look like in the "game-language".
A bell-type inequality says something about expectation values of measurements in pre-defined bases.
The equivalent game dictates that players receive a random question, and depending on the question, they measure in a particular basis.
Now, you will find that the number of "expectation values" and the "number of questions" are the exact same. Moreover, the corresponding measurements turn out to be along the same basis (assuming we start from the same shared state)! Thus, when performing an optimal stategy multiple times, the players of a pseudo-telepathy game basically sample the expectation values.
The tricky part is that the expectation values in Bell-inequalities do not correspond to winning probablities $\omega$ (a number between 0 and 1), but to so-called "biasses" $\varepsilon$, which is simply $\omega$ rescaled to lie between (-1,1) (indeed, the values of a quantum measurement outcome!).
$$\varepsilon = 2 \omega - 1$$
Take for example the CHSH game, where the optimal winning probabilitiy $\omega$ equals $\cos^2(\pi/8) = 1/2 + 1/\sqrt{8}$ (on average, per question). Turning these into biasses, and adding all 4 questions together, gives the well-known bound $2\sqrt{2}$, which is indeed the maximal violation of the Bell-inequality.
Now, you can simply look at any Bell-inequality experiment, turn it into the "game-language", and see how well the "players" scored, on average. For example, in the 3-qubit Mermin inequalities paper, take a look at figure 2. It shows the "biasses" per question (a,b,c). The "worst" expectation found was −0.655 where they should've measured -1, which means a bias of +0.655. This corresponds to a "winning probability" of 0.8275 on that question. Considering that this was the question on which they performed the worst, they clearly did better than the 0.75 classical bound.
