If he would have given it gas and rode out the slide instead of slamming on the brakes he probably would have been ok.
This could start a long derail (sigh) but what you've said is questionable . . . mainly due to physics being physics. I had this similar conversation a few years ago too. So been over this before.
The weight and momentum of the car is what it is and he had too much speed going into the corner. Having made that mistake there was little he could do to fix it. {Back to this later}
The turning force that a car's wheels can supply is only going to be so much. The opitimum grip and traction is from a wheel turning with a contact speed matching the road surface, ie not locked up and also not spinning faster.
This is seen in motorsport ( except drifting and rally ) where the driver's efforts are to get maximum power out of the engine ( or brakes ) without breaking traction. If you've seen a drag race in which one guy gives it a little too much throttle off the line, spinning the wheels, he instantly looses most of his forward speed. The opponent car will jump forward in comparison. In a F1 race you will never see a tire smoke up unless a driver makes a mistake from pushing too hard. During the 90's - mid 00's there was traction control and ABS braking.
Now with drifting the effect of a high revving engine, tyre smoke and perceived car speed is that these cars are ripping around the corners massively quick. It is all quick and the added tyre spinning & using the powerband of the engine adds a whole new dynamic to the car control/handling. But if the same car were to drive around the corners while keeping traction it would take the corners faster.
SO back to the guy in the Viper in the GIF & video. He did need to slow down, so jumping on the brakes, yes. But the brakes did lock up for a second or two, something he didn't need. But also he should have turned earlier, even crossing to the wrong side of the road. To have any hope of saving the car. He actually could have given a quick blast of throttle here (as in what you'd said) to get the car into a more aggresive turning angle as quick as possible & remove understeer, at corner entrance. But still needed to take the corner while maintaining traction.
Once he got the car off the road onto the dusty sandy looking dirt, he was gone.
Oh, and pic for the cause: