> Bitcoin’s block interval is ten minutes so it takes about five minutes on average for a new transaction to find its way into a block
is wrong. The block interval is ten minutes on average, but not ten minutes uniformly. And in particular because hashing works via random trials whose probabilities of finding a sufficiently long zeroed prefix are constant, the amount of time since the last block is no information about how much more time remains until the next one. It's always ten minutes away from now, on average, which means the average transaction takes ten minutes, not five.
The error here was in multiplying the average block interval (10 minutes) by the average fraction of that interval remaining when transactions occur (0.5). But the first operand is only valid if you assume the average block interval weighted over all transactions is the same as the average block interval weighted over all blocks, however transactions are over-represented among blocks that take longer: on average a block which took 15 minutes will have three times as many transactions as one which took 5.