Rockets generally want their fuel to be sitting on the bottom of the tank, where the engines are. That's easy enough when the rocket is sitting on the ground and when the engines are firing, but once the rocket starts coasting/decelerating (e.g., when the engines turn off due to reaching the end of their burn) the fuel may drift away from the fuel intakes, resulting in the engines ingesting vapor/gas the next time the engines turn on. Rocket engines are designed with a pretty specific operating environment in mind, so ingesting vapor/gas instead of fuel usually leads to the engines expressing their displeasure in a very vocal fashion.
This poses a challenge for staging. The naive way to stage is to turn off the previous stage's engines then ignite the next stage's, but the time between the first set of engines turning off and the second set of engines reaching a sufficient thrust level to keep the fuel at the bottom of the tanks may be enough for the fuel to drift away from the fuel intakes, especially if staging occurs lower in the atmosphere or after an extended coast period.
One way of addressing this issue is to use "ullage thrusters" - small rockets that maintain a small amount of forwards acceleration during staging to keep the fuel at the bottom of the tanks. This is what the Saturn rocket did between the first/second stages.
Another way is to "hot stage" - ignite the new set of engines before the old ones cut out. This is what the Soyuz does (and is why its stages are connected with a lattice - to let the exhaust out), and is what Starship was trying out this time. This can be simpler than using ullage motors since there are fewer pieces, but also poses some additional challenges in that the first stage needs to survive the second stage's exhaust for long enough.
The last way is to use RCS thrusters for a period to settle the fuel. This was used by the Saturn third stage before trans-lunar injection, but can really only be used once you're in orbit.
By always accelerating (with engines lit), hot staging improves the payload to orbit about 10%. So it's well worth doing.
I think it'd be interesting to see a breakdown of that 10% improvement number. Hot staging in and of itself resulting in a 10% additional payload capacity seems large enough that I feel it's rather odd that it isn't more commonly used. I'm curious how much of it is due to "direct" improvements from reduced gravity losses and how much is due to "indirect" improvements like (maybe?) not needing to save as much fuel for boostback.
A comparison against what a "Saturn-style" staging that uses ullage motors might achieve could make for a fun addition as well.
Also, If you are not reusable you have much more margin to play with. SpaceX is optimizing this thing to an incredible amount. Liftoff thrust is 2x as much as Saturn V and they are aiming at 3x as much. Total payload to orbit is 2-3x larger while being reusable.
SpaceX could have done what they did on Falcon 9 (200+ successful launches in a row) but Hot-Staging like the Soyuz is also successful.
SpaceX optimizes for long term performance and operational simplicity. Ironically that leads to more Soviet way. The N1 would also have used hot staging.
Hot firing simplifies things in this regard.
A reusable approach would involve some form of gas thruster, so might as well just try hot staging.
https://space.stackexchange.com/questions/64713/how-is-hot-s...
The primary reason for hot staging is that without it, they had to turn off all engines, mechanically "push" the ship away from the booster and then light the ship engines. Say this takes 5 or 10 seconds. During this time, the entire system is not accelerating anymore, but it's still screaming away from the launch site (because to go to orbit, you don't just go "up", you mostly go "sideways", that's why you often hear "vehicle pitching down range" during broadcasts) and to get the booster back to the launch site you then have to spend more fuel to get back to the launch site - you have to counteract all that time you spend moving away from the launch site.
With hot staging you get to the velocity you need to get the first stage to orbit sooner, because the system never stops "pushing", and there's less distance you need to cover to get the first stage back to the launch site. With means you need less fuel that remains in the first stage after stage separation, which means you can use more fuel during the first stage firing (and therefore put more mass to orbit).