"Typically" would be filling up their mined block with the highest fee transactions as much as possible before the next one is mined so that falls under your "greedy algorithm". They can maybe accommodate some requests in including TXs with lower fees but that's definitely not their priority. They are here for the money after all - let's keep it real.
Yup. Highest total fees after filling up the blocks is probably how they go, rather than just blind highest fees (e.g. if you can fit 11 txs with average fee of 10.1 sat/byte rather than just 10 txs with 11 sat/byte, that's what they'd go).
You can't really blame "greedy" algorithm, OP. It's a cost-revenue thing, quite literally.
Obviously, you can still see low fee txs, I suspect that's paid for in different ways (e.g. viabtc's paid accelerator... as opposed to broadcasting, which doesn't... accelerate).