what's key is that the energy of the system cannot change

What system? I assume you're talking about the gas surrounding the vocal cords. In that case, the energy in the system is that which is lost by the vocal cords as they push around the gas. For a lighter gas, there's less energy dissipation - the limit of this example would be a vacuum, where there's no gas at all to dissipate energy from the vocal cords. The other limit would be a fluid so thick that it's essentially solid, meaning the vocal cords barely move and all their energy goes out into the medium. So yes, the energy of the sound waves can absolutely change here.

To think about it another way, imagine moving your hand back and forth through honey - it would take a lot of energy because honey is very viscous. Now imagine moving your hand back and forth at the same frequency and amplitude but in water - much easier, i.e. much less energy is required to make it happen. A heavier gas (air) is more viscous than a lighter gas (helium). If your vocal cords are moving at the same frequency and amplitude (they are), there's going to be less energy dissipated in the helium.

So you're right that the wavelength changes. But it's not correct to say that the frequency changes proportionally because the assumption that energy is constant is wrong.

The reason helium makes voices sound different has to do with resonant waves in the voice box. The geometry of the voice box doesn't change when you put helium in it, which means that the resonant wavelengths don't change either. So looking back at that one equation, f = c / L, the wavelength L hasn't changed. The speed of sound c has changed - it's higher in helium, giving you a higher frequency for the resonant waves in the voice box. As those waves travel out of the voice box and into ambient air, their wavelength will change but their frequencies will still be higher. And that's why helium makes voices sound higher.

I'm explaining this from a physics point of view (I'm an engineer), but sound people call this timbre. It's the reason that two instruments can play the same note but sound completely different. Explaining it more than that is beyond my expertise and beyond the scope of this ELI5.