First of all, grabbing a budgie and forcing it to be near you is not going to tame it.
you're budgies are afraid of you.
To get them used to you, you need to sit by the cage, talk to them, place your hand inside the cage WITHOUT touching them, so they get used to you sitting there, and your hand being inside their cage.
Taming a budgie takes time and patience.
They need to learn your not going to hurt them, budgies do not like to be grabbed and this will really scare them and make them not trust you.
please try taking slowler steps and being patient.
Like I said just place your hand inside the cage, eventually they will stop flapping around and just watch you.
when they are used to you placing your hand in the cage, try offering them millet while you are holding this.
I hope the budgie is okay, I don't really understand what you did to it.
And No you should not get an african gey or a cockatiel, they will be no different if you don't have patience to tame them! and bigger birds like african greys need LOTS of care and patience from their owners.
I hope you have more luck with your budgies.. please be patient with them, when they bite they are afraid they not trying to hurt you.