This is correct, the power density of electric motors (and the batteries to power them) is still very far short of what you can get from a well-designed turbine-based pump (that is, a turbopump), but there's so much ancillary plumbing and and stuff for a turbopump (like the precombustion chamber that has to generate some hot gas to drive the turbine), that doesn't scale down all that well to the small scale of the engines on the Electron, that the power density advantages of a turbopump tend to asymptotically tail off. Also, they require a lot of expertise to design, compared to almost any other kind of pump. There are turbopumps and turbopumps, though, and here I will paste in an HN comment of mine from a number of years ago:
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There are three kinds of rocket engine cycle (well, there are maybe more but these are the three that have been flown historically). The Expander Cycle, the Staged Combustion Cycle, and the Gas Generator cycle. I'll mention the last two.
Merlin, as the article mentions, is an example of a Gas Generator cycle. In this cycle, you take off a little bit of fuel and oxidiser to burn outside the main combustion chamber, to generate some hot energetic gases that you can exhaust over a turbine. This spins the turbine up, which is connected to a shaft with a compressor on the other end. The compressor increases the pressure of the propellents so that they can be injected into the main combustion chamber. This assembly (turbine, shaft, compressor) is called the turbopump. It's necessary because the engines require very high flow rates to get the thrust they need, and that has to be at a high pressure - higher than the pressure of the combusting gases inside the combustion chamber, else you wouldn't be able to inject it!
Back to the bleed-off to drive the turbine. You usually don't want a perfect stoichiometric mix of fuel and oxidiser for this, or even close, because it generates extraordinary hot gases that no turbine would last long in (The turbines are spinning at many tens of thousands of RPM usually so would be subject to much higher forces than the actively cooled walls of the main combustion chamber). For this reason you usually have a large imbalance of one propellent to the other to keep the temperature down. Usually you run with excess fuel, or 'fuel-rich', as the opposite - oxidiser rich - means you have hot oxidising gases which are harder on the metallurgy. I do know of some russian exceptions to this, though, where fuel rich would have left sooty deposits in the plumbing (The materials science employed in the turbines was apparently so witchcraft that when the US got intelligence of oxidiser-rich turbine precombustors, they thought is was deliberate counterintelligence from the russians to get them to waste billions researching the impossible). The gas generator cycle, as the article mentions, dumps this turbine exhaust overboard separately. The problem with this is that there's a load of uncombusted fuel in this exhaust, which you're just wasting, and this hits your rocket performance - the Specific Impulse ( I_{sp} ), as you're not getting as much bang out of a given mass of fuel as you could.
The answer to this is the Staged Combustion Cycle, where you also inject the exhaust of the turbine into the combustion chamber to finish off combustion. The performance of these engines is higher but the thermodynamic balance to design a working system is a greater challenge, and some of the engineering is a bit harder too. Staged Combustion engines are mostly russian, although the Space Shuttle Main Engines are a US-design example of Staged combustion.
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Staged combustion engines are extremely efficient and on a big engine no electric pump system will even touch them, unless there is some materials-science breakthrough that will allow us one or two orders of magnitude improvement in flex density in electromagnetic materials. Electric pumps will probably remain in their niche for small engines and satellites.
I do think that small turbopumps are worth further research, although I don't know if the market needs higher performance small engines over more cheaper-to-produce small engines, but certainly there was fascinating work done in the uk in the 70s with tiny turbopumps (about the size of a coke can) that ran at hundreds of thousands of rpm, with a power of megawatts, and compressors very cleverly shaped to run sustainably far beyond the cavitation point of the fluids, which is usually the point at which you can't pump anymore, in traditional pump design literature. In combination with an expander cycle you could probably produce some extremely high performance, simple, small rocket engines. Maybe.
We live in exciting times.
