if someone generated 1000 transactions every minute at zero fee.
Would it mean that anyone else who tried bitcoin would have very little chance of getting confirmed?
The chance of such transactions getting confirmed is very low of less than 1% Earlier when the network isn't that congested there were transactions that gets confirmed after a long time when no fee is provided. Now the miners just reject the transaction, so no transaction is being confirmed without fee. Even the fee provided based on the transaction volume is getting stuck to the network out of congestion.