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My understanding (and I'm no authority in this subject) is that adding lead to gasoline allows achieving better octane ratings (ie lower chance of self-ignition). This reduces engine knock, which makes the engine live longer, also allows pushing more fuel into a cylinder per ignition cycle more confidently.

These things are possible without lead as well, it just costs more money to do so.

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It's about increasing the compression ratio without self-ignition. High compression ratios mean higher power for the same weight, and airplanes are all about minimizing weight. You also get higher efficiency. Again, lower fuel weight = good.

You can dump as much fuel in the cylinder as you want (in fact, most airplane operating handbooks call for a very rich mixture on takeoff because of the cooling effect).

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