Or you can lose him on day 2, or day 3, or day 4.
Reveal the D and they'll know he'll be protected, so they go for someone else. There's more than a 1 in 3 chance that it will be one of the BG or jailer in the first 2 days. If one of them are killed, the D is gone next day or day after. Meanwhile, the only benefit the Cs have got in 1 G and after that 1st day, there will be a lot of fruitless looking until the D gets killed and the Cs will be a lost flock lynching randomly.
Take tonight for example. Lets say that you were the D and revealed to the rest that I was an investigated G. I was gonna do everything to put doubts on you and that was gonna be my only chance to save my ass. But I was probably gonna fail. The jailer puts you in jail. The BG protects himself.
The Gs know that the D and the BG are protected so they fish for the jailer. They have a 1:11 chance to get him (out of the 12 civilians they only know the D). That's small chance, isn't it?
But lets say they got super lucky and got the jailer. A new jailer gets that role.
On the 2nd day the BG protects you.
Now they again need to fish for the new jailer, not for the BG.
They have a 1:10 chance to get him today.
If they succeed then you get a 3rd day of investigation, which is quite enough.
But in order to succeed they will need to beat twice in a row an 1:11 and 1:10 chances.
Do you think that's possible?
If they fail only once, the D gets a 4th day of investigation. 4 investigations are enough to win a game, even if in 3 of them he gets a C because he will already know those 3 investigated Cs, he will know the BG and he will know the jailer.