First of all, Bitcoin block rewards won't suddenly stop because it's not a centralized system which is under the control of an entity that can decide when they are going to stop the block rewards, it's a system that is automated and decentralized, and the rewards will only stop once all the bitcoins are mined and in circulation, and that is estimated to happen in the year 2140. Before that, only halvings will happen, which will cut the rewards in half after every four years, which will keep reducing the block rewards for miners.

Once all the bitcoins are mined, miners will still be getting rewards through transaction fees, and transaction fees aren't that low for every block, miners can sometime get good rewards only from the transaction fees even if they are not getting anything as block reward. Besides, who knows, by the time block rewards are finished, Bitcoin might have a very high value, and even the fees might be enough for miners.