When the last bitcoin is mined/bought, it would be harder for everyone to facilitate transactions more so afford them because the system will be overloaded and as a countermeasure, will charge higher fees and would slow the process down for the rest in queue. The thing about POS is it ensures efficiency of service and the price of each transaction which in theory should work, especially for coins with high potentials for scalability like bitcoin and ethereum, with these two being the most common victims of the shortcomings POW has.
Blockchain has trilemma in terms of scalability, security and decentralisation. If we talk about PoW, then it has performed well under the umbrella of BTC. It has scalability issues but its secure and ensures decentralisation of btc. While we have yet to see PoS working under a decentralized coin like Ethereum.
After the last btc will be mined, there will be a fee that will be paid to miners as reward. Like I said PoW has scalability issues but still with this issue btc manage to grab over 1 trillion usd cap. I m very much sure PoW can manage the load after last btc is mined as its tested under extreme load.