I remember reading years ago, before Falcon Heavy actually launched, that they'd considered running all engines with just the fuel from the booster rockets, and once they were empty they'd release and have a completely full core stage. They don't actually do this but I don't know why its better.
It works GREAT in Kerbal Space Program! They call it "aspargus staging, because you can have a rocket with say a core and six side boosters just like the core, have the whole combo of 7 using just the fuel from two outer boosters, they run empty quickly sep, then use fuel from the next two, and so on.
I did many rockets in KSP using that method.
SpaceX did indeed plan to do that, but it is hard to do in real life for various reasons. From:
https://en.wikipedia.org/wiki/Falcon_Heavy#Propellant_crossfeed
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"Falcon Heavy was originally designed with a unique "propellant crossfeed" capability, whereby the center core engines would be supplied with fuel and oxidizer from the two side cores until their
separation.
[84] Operating all engines at full thrust from launch, with fuel supplied mainly from the side boosters, would deplete the side boosters sooner, allowing their earlier separation to reduce the mass being accelerated. This would leave most of the center core propellant available after booster separation.
[85]
Musk stated in 2016 that crossfeed would not be implemented.
[86] Instead, the center booster throttles down shortly after liftoff to conserve fuel, and resumes full thrust after the side boosters have separated.
[3]"
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Note that Delta-IV Heavy did a sort of similar thing, throttle back the core, saving fuel in it, then dropping the outer boosters when they ran out, and thrusting onward with the center core till it ran out of the fuel that had bene saved during the low throttle phase.
Note that for on FH, all 27 engines are ignited at liftoff, core is not "air started" like Titan-III/IV. If the Merlin engines did not have as low of a throttle capability, in theory some engines could have been shut down early, then re-ignited shortly before staging, since those engines have re-ignition capability (only times any have failed to ignite, have been when they ran out of the re-ignition fluids - TEA and TEB, and future boosters were modified to have larger storage tanks for that). But they do have enough throttle to not need to do that.