If a bank client really sees a bank employee gambling during working hours, and even at the workplace, then this will clearly make a negative impression of the bank as a whole. And if the employee really was warned by the employer, but he did not give up such behavior and his gambling, then of course the employer must fulfill the promise and fire him.
Of course, you are also right that dismissal follows warnings and in the event that the employee did not listen to them.
And the second option is image dismissals, which follow immediately after some obvious negative behavior of the employee. For example, he commits a crime and the employer finds out about it. Naturally, the bank cannot afford to keep employees with a dubious reputation.
In theory, how can a bank client see what employee is doing at work, when they sit on different sides of the table. If employee turns his monitor, and while having a conversation or giving explanation, gambles parallel on banks PC, then that is unacceptable. But would an employee really be so stupid to do that? You are absolutely correct that bank should not keep an employee with dubious reputation, but bank should not hire such person at first place. If they really had such an employee, then its HR department fault for giving a green light when they hired him.