What, other that Fees and Luck, controls the time to confirmation? Are there any factors that contribute to how fast a transaction is confirmed other than the Fee (sat/B), and a little bit of luck?

Does it matter what service is used?
How large the transaction is?
How many senders/receivers?
Which transactions make it into a block are purely at the discretion of the miner who mines that block.  In general, they take the transactions that will yield them the most in fees...but there's not really a way of saying for sure.  If a particular mining pool has another incentive in play that is worth more to them than the fees they'd be sacrificing, I'm sure they would operate differently.