And even if I went your way, it would only make the mystery much deeper: If really most empty blocks are found within a second, and we have about 1% empty blocks on average - that would mean the probability to find an empty block within 1 second is 1%. Not 1/600th of the roughly 50% at average block time of 600s.
Do you imply that different mathematics of difficulty apply to empty blocks?
If yes, why would any miner "waste" 99% of his computing time on finding full blocks, when mining empty is so much easier?
The way the difficulty works should be the same as well. If it's empty, the merkle root is the TXID of coinbase so there is really no difference in the difficulty whether you are mining empty blocks or not. You could use covert ASICboost and mine empty blocks but that probably wouldn't make sense economically.
I see that you understand that the distribution of the block intervals follows an exponential function and that the probability of a block being 1 second from each other is 0.167%. The problem with this is that there is no perfect information when you're looking at block intervals. Propagation of the transaction across the network takes some time (depending on how well your node is connected). The only metric that is remotely usable is the timestamp that the block is received by the node which is subjected to the delay as above.
It would be a stretch to assume that all of the blocks are mined within 1 second of each other. It's much faster to just take just the hash of the block header on the other pool than waiting for it to be relayed through the network.