Yes, totally doable. No third party app needed. Just use the “pipeline automation” function. One of the options is to “move this deal” when it enters a stage in the pipeline. We use it all the time now. It’s wonderful.
Basically if you look in Easy Automations there is a Move Deal under then which can move from Any Stage Any Pipeline to anyother.
The main thing you have to think about is the trigger. You could have a Stage in your original pipeline that is Move to Bob, Move to Mary, etc and these could then trigger the move to the different sales team pipelines. That could generate a lot of stages in your initial pipeline however if you have a large sales team.
If you combine this with advance automations you can make it more seamless and only need 1 step, however thats beyond a simple text answer to be honest, drop me a PM if you want to discuss it more.