I gave you an example that only uses the benefit of mana crystal and tried to show most situations in that it is still helpful. Your example with the lair uses multiple cards, and multiple sources of channeling. Harmonize is almost identical to mana crystal except that it can target something other than the mage. So your example did not exemplify the usefulness of lair but rather the usefulness of lair+harmonize (or even mana crystal with a similar effect)+ring of beasts. My example judges mana crystal on it's own terms. I could further the effects of mana crystal by attaching a ring or gate of voltari or anything else that in creases channeling but then I am not judging the value of a single mana crystal
My point with the Lair (of, if you prefer, with Lair + Harmonize) is that you can achieve something with that strategy that is impossible without it, namely having 4 foxes, 1 ring of beasts, and moving 4 zones by round 3. There is no other card (or combination; whether you consider the lair alone or with harmonize is irrelevant) that will allow you to do that, so we can conclude that the lair has a unique payoff by the 3rd round of play.
None of your examples with mana crystal demonstrate a unique payoff before the 6th round because it is always possible to achieve at least the same game state (ignoring the physical crystal itself) without using a mana crystal. To look at your example:
With crystal:
Turn 1 (10): Crystal + Caltrops + Move Once (0)
Turn 2 (11) : Dwarf (0)
Turn 3 (11) : Dwarf (0)
Without crystal:
Turn 1 (10): Mangler caltrops + Move twice (5)
Turn 2 (15): Dwarf + Enchanter's Ring (2)
Turn 3 (12): Dwarf + Arbitrary Enchantment (0)
At turn 3, the crystal user is 1 move, 1 enchanter's ring, and 1 enchantment behind the non-crystal user, and both have the same stored mana. There is no unique benefit to having the crystal by turn 3.
If we look at whether there is a benefit by turn 4:
Crystal
Turn 1 (10): Crystal + Caltrops + Move Once (0)
Turn 2 (11) : Dwarf (0)
Turn 3 (11) : Dwarf (0)
Turn 4 (11) : Dwarf (0)
Non-Crystal
Turn 1 (10): Mangler caltrops + Move twice (5)
Turn 2 (15): Dwarf (4)
Turn 3 (14): Dwarf (3)
Turn 4 (13): Dwarf + Enchanter's ring (0)
Both players have the same number of dwarves and mana, but the non-crystal player has 1 move and 1 enchanter's ring over the crystal player. And if we consider 5 turns:
Crystal
Turn 1 (10): Crystal + Caltrops + Move Once (0)
Turn 2 (11) : Dwarf (0)
Turn 3 (11) : Dwarf (0)
Turn 4 (11) : Dwarf (0)
Turn 5 (11) : Dwarf (0)
Non-Crystal
Turn 1 (10): Mangler caltrops + Move twice (5)
Turn 2 (15): Dwarf (4)
Turn 3 (14): Dwarf (3)
Turn 4 (13): Dwarf (2)
Turn 5 (12): Dwarf (1)
Now they have the same number of dwarves, but the non-crystal player has 1 move and 1 mana over the crystal player. By Turn 6, of course, the crystal has paid for itself, but the non-crystal player still has the advantage of 1 move.
My point is this: If, without a crystal, I can achieve everything by turn X that can be achieved with a crystal, then it is incorrect to say that the crystal provides a benefit by turn X. The examples you have given so far fail this test because in each case it is possible to achieve the same thing (and more) without needing a mana crystal. If you can show me an example where you can reach a game state that is impossible without a crystal, then I will be convinced; otherwise, I still am certain that the crystal has no net benefit until 5 turns after it is cast.