I do not have Privet in Florida.
I have Yaupon Holly, about the same size when immature.
Ratchet Rake will tear similar spreading stuff out by the roots, but if Privet spreads via roots, small residual bits remaining in the soil will still need spot herbicide application to kill 100%.
So, if the RR will pull out 95% of the plant, that still not is enough to eradicate the plant without some herbicide spot application.
2, 4-D based herbicide is probably what you need to complete Privet kill, rather than Roundup/Glycophosphate.
I have also had good success with Gordon's BRUSH KILLER from Tractor Supply Company, which only requires 2-1/2 tablespoons of concentrate in a gallon sprayer, so a half gallon of concentrate lasts a long time.