There are seed. All the distributions are known for the front admin. So why not do this?
Example.
GAME #667480-GAME #667486
cd11d221877f686fa16d95ca749838ca0df4e8eed5d52f20eba632520b26c9bf
9ac1a7603e867723875a708a94db81d5377458572cd35f8675f3351eb92aad5d
267bf1cc09f39bc03a3156be3bbba646dcb1921f5ce6a88f130b6cd70c2d7d1f
554aca7806e32edde232b863543722191756e59c5f2b32c27d7ec3ba4cc8585c
31ece5a9e0c34995425b4a2a44208b2d8118712bd27688d39c2939778c506aeb
15441a853f9b82bba7cdcea821d6ac96104ef272ab42f90cbcfd16ce033ea8a4
220e10106ab97bcf2173dd2e50568bbcd4c98157da1041fdcfc0c5b59013d8a8
Result SHA256
26a3597e1f7a17034307a51ee8b7a30193519e2e141b8dfcce79679d9dc1f764
Your proposed system would allow you to batch verify that N games were predetermined. However it besides the usability problems (you need to collect the result hash, then collect all N games hashes before you can verify) it also suffers a critical flaw: it doesn't prove the hashes are drawn from a fair distribution. If your scheme was used, Daniel could just pick all bad hashes and then commit to them.
As scant says, the hash chain method is far more elegant and offers all the guarantees yours does, plus more.