Why not give base_fee to current block miner?
If the miner of the block received the base_fee, then there would be effectively no change with EIP-1559. Miners would still be able to include their own transactions at no cost to fill a block. Miners would be able to offer rebates to users for transaction fees via secondary channels, thus allowing them to effectively include transactions priced below the base_fee. See section 5.3 of http://timroughgarden.org/papers/eip1559.pdf for a much more technically deep answer.
If miner fees from transactions for a block relative to the block reward are high, selfish mining attack vectors open up (see: https://www.cs.princeton.edu/research/techreps/TR-983-16). By not giving the base_fee to the block miner, we mostly (but not fully) mitigate this attack vector and greatly mute the problems this vector has on the blockchain.
Why not give base_fee to future/past block miners?
The security needs of the blockchain don't coorelate with block space demand. Ethereum doesn't need more security during times when block space demand is high, and thus fees are high (like late 2020). Inversely, it doesn't need less security in times when block space demand is low (like early 2020).
If it is decided that we need more security, we can increase the block reward which is slightly lower complexity and doesn't result in a variable level of security based on block space demand. A block reward increase also allows miners to more reasonably reason about their ROI due to having a stable block reward.
Since block space demand is not something we can control, it tends to be very volatile. It has a high amount of seasonality (daily, weekly, yearly variance) and it is very sensitive to external events that can result in significant changes in demand over time. In order to have stable network security, we want the amount we pay security providers to be stable. By giving miners transaction fees, we are creating an instability in the network security budget, which results in an instability in security.