I think Bronn inherited the role due to his reigning over Highgarden and the Reach. Yes, the Lannister army sacked Highgarden for all of the gold stored there. But it is the capital of the most fertile and prosperous region of Westeros that remains, since the gold mines near Casterly Rock have run out. The crown would need the support of the Reach, and a good way of ensuring that support and loyalty would be installing it's lord as Master of Coin (and most likely Warden of the South).
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