Should a soldier be switched units upon promotion to E-5?

Absolutely not. You'll be dealing with your peers being placed above you, or being placed over your peers, your whole life. There's no reason to try to pretend it's not an issue by avoiding the conflict. The sooner a new NCO can overcome that first obstacle their authority, the sooner they can grow as a leader. It's better to face that early in your career and understand how to deal with it than it is to face it for the first time as a senior NCO and not know how to navigate that thorny issue. As you get promoted higher and higher, the promotions slow down and suddenly you're all the same rank with people who out ranked you years ago, and people you out ranked as well. You'll be placed on projects where one of you is in charge of the others and you have to know how to lead them

I only realized it later watching the series....

Depends on if there is an opening for him, or her within their MOS, or they need to retrain for a different MOS.