safely introducing budgies

i have 2 budgies in my flock with mango, and both of their cages have been in the same room for almost 2 weeks. when the budgies come out they usually fly onto mangos built in playgym on the top of her cage, sometimes shes up there too and others shes inside.

if they get to close to her she kinda looks like she wants to go after them but i never let her. and once when she was inside the cage and one budgie was ontop of her on the bars she tried to pull a tail feather out lol but the budgie flew away.

im thinking this is probably a territorial thing, cuz its mangos cage. ill probably get the birds a playgym for christmas to share together as neutral ground. is this the safest way for them to play together? obviously ill watch them every second to make sure she doesnt chomp one of my lil babies.

im hoping they can all be playmates like some others have. of course it depends on the birds personalities and if they click or despise each other. any advice would be highly appreciated

I think your on the right track with a seperate playgym just make sure to put them all on it at same time so no one claims top spot