What will happen when only large institutions can afford to use Bitcoin's blockchain, due to rising fees?
I don't think only large institutions can use it. Everyone can use Bitcoin although they must pay higher fees for the transaction now. You must know it is not about large institution or small institution, all of people pay higher fees now. When you have a large number of Bitcoin, you will pay bigger fees as well.
By the way, the high transfer fees won't last forever. It will gradually decrease again when the number of transactions will be decreasing. If it is getting higher now, it is because more people trying to move their Bitcoin to CEX. They may prepare to sell their Bitcoin, they may try to rebuy when it is dropping again the price.
Won't this centralize the network if most can only trade bitcoin on centralized exchanges?
Bitcoin can be traded in decentralized exchanges. You are wrong if you assume it is only traded in centralized exchanges.
We have many decentralized exchanges, some of them are even popular enough.
Take a look on this
https://coinmarketcap.com/rankings/exchanges/dex/.