I can see one crucial difference in the gas limit and block size limit:
Block size is constant
If you increase it - you increase it for everyone. Gas, on the other hand, is the situation that participants can pay for more resources.
The increased/flexible bitcoin block size seems to have few interesting implications.
With constant block size the bitcoin supply is constant and steady
If you read original Satoshi's paper - it seems like by design. Because the block size is limited, and the difficulty is adjusted so one block is mined every 10 minutes - you have steady increase +12.5BTC in the network per minute. (It will be halved and halved until there's 21mln BTC in the network)
What could happen with "infinite" block size?
It sometimes helpfull to analyze the extremes, so let's do it. If block size limit does not exists, a miner can process all transactions in the queue. One of the result would be
every transaction is processed.
So an adversary could spam the network (https://bitcoin.stackexchange.com/questions/16942/is-it-possible-to-spam-the-bitcoin-network-with-dust) to the point the blockchain grows to the size it can not be supported by any device (https://www.reddit.com/r/Bitcoin/comments/7oszqh/eli5_why_would_bigger_blocks_lead_to/). And BTC is dead.
Block size is flexible, and miner pays a fee for bigger mined block size?
First problem would be - to whom miner pays the fee? Would it be proportional refund to every sender? Does it mean more transactions to mine?
I'm not sure if the protocol allows anything like this, but let's assume it does, and miners have to balance their profit as
fees_from_transactions - big_block_size_fee
This basically would be ETH approach - transaction sender can pay bigger fee to offset big block size cost.
Except of the first problem mentioned - I can't see a problem with that really.
Last extreme - allow more blocks that 1 per 10 min
I'm not sure how this could be achieved, but probably by setting the difficulty to a constant value.
Con of this approach would be that in periods where there are less transactions: it could lead to new block every few seconds with one transaction in the block. That would bloat the blockchain size with a lot of "metadata".
So, in my opinion, it's a tradeoff game, and we would have to see how different cryptos take different paths and how it pans out.