There have many instances where the ISS only has 3 crew members, often for many weeks. This places significant strain on workload and productivity. Why isn't a new crew launched quicker?
There are probably several reasons and it is a mix of them all.
At some levels your question is somewhat specific on why they do indirect handoffs, where one crew of three remains, one crew of three departs, and then later one crew of three joins the station.
On Mir with only a single active crew (for the most part) they would do direct handoffs, where the second Soyuz would dock with three new crew members and co-habitate for a few days to a week and then the old crew would take the old Soyuz back.
Sometimes for long duration missions they would launch a fresh Soyuz with a crew (Often selling a seat to foreign nations or a tourist) who would stay for a few days to a week and return in the old Soyuz, giving the active crew another 200+ days of life on the Soyuz.
Usually this is a decision based on scheduling issues. There are many components.
The Soyuz vehicle itself has a lifetime in space. (Dragon Crew and Starliner will also have similar lifetimes). As they get close to that date, they need to leave.
Rarely do they go right to the wire, since you never know when an emergency could come up and you might need some extra time before leaving.
You need to have a new vehicle and crew available to launch, which has its own basket of scheduling problems. (actual hardware, launch site, launch window, good time to dock vs bad times to dock). You might be waiting for some piece of equipment to bring up. There might be other ops in progress that interfere on the station.
In some cases it is a cost savings measure. Need fewer Progress launches of food/water/supplies for a crew of three vs six. Need fewer Soyuz vehicles.
There was a point where I think the crew on ISS was down to 2, the bare minimum for that reason.
On the US side, this is going to go to 4 per launch and the Russian side will continue to launch 3.