If you "remove fees", and I contact a mining pool and offer to send them a payment to confirm my transaction in their next block, how are you going to prevent me from doing that?
I like that method better. Basically you would have to be lucky enough to pay the right pool. It is a lottery whether the person you paid actually wins the block or not. This would effectively set fees to zero for normal transactions and if someone really needs something expedited then they can try their luck on guessing who the next person to win the block will be.
And if I start a business that contracts with MANY of the largest mining pools? Then my business can charge users to "accelerate" their transactions. My business can just pay whichever mining pool (or solo miner) includes the transactions from my business (after they broadcast their block). I can let each user set their own price that they want to pay for their transaction, and then I can sort them for the pools and miners based on which transactions pay the highest fees per byte.
Anyone that fails to use my business (or any other similar business) will never see their "free" (or low fee) transactions get confirmed. Instead they will just see a bunch of transactions that appear to be free (but waiting less time than them) getting confirmed instead of them.
In this scenario, I now have a LOT of influence over the system. People can't get their transactions confirmed if I don't want them to, and they don't know how much of a fee they need.
This is just a MUCH worse fee system than the current system. No thanks. I prefer the transparent fee model where everyone can see what all the other transactions are doing.