Rocketlab's Electron uses an electric pump combustion cycle with no preburner so all fuel is fed into the combustion chamber rather than wasted in a preburner/staged combustion-powered turbopump that requires complex engineering and finicky systems so my question is why do/are larger new rockets not using electric pump cycles? With improving battery and electric motor technology I would think that the fuel efficiency, throttle ability, and ease of engineering would be desirable especially with reusability being a concern now. In something like starship that will be coasting through interplanetary space, you could use the methane and oxygen boil off to power a fuel cell that would in return recharge turbopump batteries or you know you could have solar panels powered by the sun. Is weight really the only factor, I mean a super complex turbopump with all the crazy pressure budget compatibility like in the Raptor can't be that much lighter?
Ultimately the pump, whether electric or combustion-turbine driven, needs a certain amount of total energy input to do its job.
Combustion reactions both deliver more total energy per mass than batteries, and deliver it faster. From an article on gas versus electric cars:
Stored energy in fuel is considerable: gasoline is the champion at 47.5 MJ/kg and 34.6 MJ/liter; ... A lithium-ion battery pack has about 0.3 MJ/kg and about 0.4 MJ/liter (Chevy VOLT). Gasoline thus has about 100 times the energy density of a lithium-ion battery. This difference in energy density is partially mitigated by the very high efficiency of an electric motor in converting energy stored in the battery to making the car move: it is typically 60-80 percent efficient. The efficiency of an internal combustion engine in converting the energy stored in gasoline to making the car move is typically 15 percent (EPA 2012).
So the mass ratios are significantly in favor of combustion for cars -- at least 35:1 including the efficiency factor (but this article is from 2012; I believe battery packs have gotten much better since then). As user3528438 points out, this doesn't account for the turbopump's oxidizer consumption (cars get their oxidizer from the air instead of carrying it), so it's probably somewhere in the 10-15:1 range instead.
If the electric pump can be made much smaller and lighter than the equivalent-power turbopump, it might close the gap a little more, but I doubt that it's a mass win for the Electron.
Your suggestion of recharging the batteries during an interplanetary coast, so they need to only be sized for the more demanding injection phase, rather than injection + descent, is interesting, though, and might improve the calculus a little bit.