My thoughts on this:
(1) Rozier will have no problem playing with or sharing the ball with Gordon Hayward. It was sharing the ball with Tatum, Jaylen, Smart, Kyrie, Hayward and M.Morris that caused problems. No one element. It was everything altogether.
(2) I expect Rozier to easily beat LaMelo out for playing time. I have no faith in LaMelo's readiness for the NBA. He was inefficient / ineffective in Australia. How the heck is going to handle the NBA?
That situation won't become a problem for years to come and that is only IF LaMelo can develop into a good player which may not happen. Rozier is only under contract for 2 more years anyway so he probably won't be around anymore by the time LaMelo (IF LaMelo) pieces his game together.
(3) Rozier is the not the type of PG you build a team around or plan long term with. He is a journey man. He is too flawed of a floor general / playmaker to become a building block for anyone. He will venture from team to team. That will be his career.
And there is nothing wrong with that. Rozier will be a valuable scoring-orientated role player at PG for teams who need such a player. He is unlikely to get such a big payday in the future but he can play on successful teams if he & his prospective new team understand his limitations and find the right for him.