I think some people are making a bigger deal out of the flag than is warranted.
Whoever is closest to the hole, should tend the flagstick: Put it in if someone wants it in, take it out AND HOLD IT if someone wants it out.
When it is that person’s turn to putt, he hands the flag to whoever has finished the hole and they tend the flag.
Absolutely zero time lost, it is only an issue if someone feels they are above tending the flag when they should. I regularly see people who hole the first putt and walk away and leave the flag on the ground. I was always taught: First in, tends the pin.