It would likely be nonrandom at that point. Maybe 3 random champions? Some other games do that.
Another idea i had while pondering this was Random Draft:
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Optional: each team could ban 5 champions from selection by the pool
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14 (seemed like a fair number) champions are selected randomly for the draft pool
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Draft proceeds as normal, with no restrictions on owned champions.
Alternatively, Balanced Random:
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A teamcomp is chosen randomly, ie 1 support, 2 ADC, 1 mage, 1 fighter, or 2 support, 2 tank, one mage
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Both teams get randomly selected champions that fill the required comp
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Players can trade champions+roles
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Reroll gives you another champion of the same role
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Overall champ winrate total (or some other metric) of each team should be kept close together