How does a battery work?

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How does a battery work?

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Please first let me explain my understanding of batteries and then my confusion. Mods if this goes against rules please don’t take it down this doesn’t make sense to me and ChatGPT isn’t helping.

A battery has a negative full of electrons anode and a positive lacking in electrons cathode.
In between is an electrolyte.
When a battery is connected to an external circuit electrons flow from the anode to the cathode and do work along the way (e.g. powering a light)
The electrolyte exists so ions can also go from Anode to Cathode.

Q: Why do these positive ions go from anode to cathode? If the cathode is positive, how would it become negative enough to draw positive ions when the anode is already negative. That would imply the anode goes from negative to positive and the cathode from positive to negative which wouldn’t make sense because why would they not just go from negative/positive to neutral.

Q: What if there was no electrolyte? Wouldn’t the electrons flow and do work anyway?

Q: Do the electrons travel at the same speed as if they didn’t have to do work along the way?

Q: How does a rechargable battery get the ions and electrons back the anode?