[PROBLEM & SOLUTION] Blocks fees to be rewarded to the next block's miner?

Hi!

I was thinking yesterday about an idea that I think could help to solve one specific problem of bitcoin. Let me know what you think!

Users decide what fee they want to spend on a tx depending on how fast they want it to be confirmed. Blocks can hold a limited amount of txs, so, miners always try to include those that paid the most satoshis/byte first. When there is network congestion (more txs being sent that can fit in a block), either you pay higher fees or your txs could take longer than expected to confirm.

The client (software, website wallets, etc) suggests the user a "recommended fee" for their txs, which is calculated, among other things, depending on the fees paid by txs included in the previous blocks.

The problem: Malicious miners could use this to increase the fee users have to pay to send txs. They could create lots of spammy txs with high fees, include them in their own block, and then the network would think you must spend higher fees to get your txs confirmed. Since miners are including those txs in their own blocks (without actually relaying them to the network so that they won't get mined by other miners), they would be receiving the fees of such txs, and it would be cost-less for them.

Due to this, miners could potentially make the fees go higher with no cost for them. They probably do it or they probably don't, we don't know. But I would sleep better at night if I knew they don't do it because they CAN'T, rather than because they are good people and they wouldn't do such thing.

The solution: One easy way to solve this could be if, when a block gets mined, the fees obtained by the block, instead of being claimed by the miner that mined THAT block, they are claimed by the miner that mines the NEXT block.

That way, malicious miners would be no longer interested in including lots of txs paying lots of fees on a block to raise the "recommended fee", because they can't be sure they will be lucky and mine the next block to get the fees they paid back. It would no longer be profitable for them.

Miners would however still be interested in filling in the blocks and including as many transactions as possible, since they could actually be the ones to mine the next block, and if that's the case, they would still want to get the biggest reward possible.

How to implement: I haven't thought this part carefully yet, but I think this could be done in a way that is completely backwards-compatible. All we would need to do is to add a new rule on future versions of the bitcoin client in which they will only validate new blocks if the coinbase tx pays the reward to an specific "anyone-can-spend" output, such as an script that doesn't require any signature to be validated, or just a bitcoin address derived from a private key which is public and hardcoded in the client. That way, miners could add a secondary transaction spending the coins gathered by the previous block (or actually, 100 blocks before, since the coinbase txs needs to mature for some blocks before it is spent).

Thanks for reading! Thoughts?