Moving to PoS is a necessary but not sufficient criteria for reducing transaction fees. The Ethereum merge will reduce them slightly (as it will reduce the block time from 16 seconds to 12 seconds, or something like that) but it won't make any big difference. In the coming year, there probably will also be general ZK-rollups on Ethereum mainnet that will reduce transaction fees substantially. (You can already use ZK-rollups for some limited things like transactions, but not for arbitrary Ethereum operations.) Also, way out in the future, there will be data sharding which will reduce fees on mainnet significantly, look up "proto-danksharding".
Transaction fees are a market-based mechanism to allocate a scarce resource (limited amount of block space on a given block chain). Moving from POW to POS will not reduce transaction fees in any appreciable amount, instead the fees previously paid to miners will instead be paid to stakeholders.
> Doesn't the block generation become faster with PoS and thereby increase the supply?
In the most technically-correct sense: No, switching out PoW with PoS does not guarantee faster block times. There can exist a PoS chain with 10 minute block times, just like there can exist PoW chains with <1 min block times.
I'm guessing that this opinion came from the early days of Ethereum, where the roadmap for the transition to PoS was still unclear. Now, as ETH's switching system is being tested on its testnets & is more or less finalized, it's been made clear that the block times will only decrease from 14 seconds to 12 seconds - A ~14% decrease, which is significant, but not by much.
> Whereas under proof-of-work, the timing of blocks is determined by the mining difficulty, in proof-of-stake, the tempo is fixed. Time in proof-of-stake Ethereum is divided into slots (12 seconds) and epochs (32 slots).
The truth of the matter is that an increase in block speed will compromise on either decentralization or security/finality: Increasing ETH's TPS will compromise on one of the two. As such, there's been a shift in focus from relying on ETH for everything, & instead towards allowing ETH to focus on security & decentralization, while the L2 solutions (Arbitrum, Optimism, zkSync, etc.) focus on scaling the TPS, utilizing/renting ETH's security for their own benefit. This way, ETH's overall ecosystem gets to scale up, while still maintaining its security & decentralization.
