You have no idea what you are talking about.
I used to work in government, and I have many close friends and colleagues who currently work in government, but I don't have a clue.
A flex day is simply a way to work the same number of hours over a slightly shorter number of days. Nothing "bonus" about it. You work the same number of hours for the same amount of pay.
It's a perk.
Let's walk through a scenario, shall we? You are offered two jobs. They are both dream jobs in your field of work. The pay is the same, the vacation is the same, the benefits are the same, and you are otherwise indifferent about working for these two particular organizations. The only difference is that Job A offers flex days while Job B does not. Which offer would you accept?
And yes, it is available in other industries where an employer wants to offer something the employee wants without having to pay any more money.
If an employee wants it, and the employer is willing to provide it, how is it not a perk? It's totally a benefit to working in government.