If you cover the air intake with your hand, effectively sealing the intake, the engine SHOULD turn over more easily because you are creating a partial vacuum on the intake stroke and the compression stroke has less air to compress, thereby doing less work. Less work per stroke, same electrical power input (almost, making some assumptions), more strokes per unit of time.
OK, so what are the possible causes? Creating that additional suction at the carb will draw more fuel from the bowl through the jets. The additional fuel, plus the faster starting speed permits starting.
I think you have a poor positive or ground connection to the starter. Clean the ground wire at both ends, then clean the ground at the battery. clean the positive leads at the starter relay and at the starter itself. That may give you enough voltage to turn over the engine faster, creating lower pressure through the carb's venturi, thus drawing more fuel.
If the carb jets are dirty, or the fuel bowl isn't filling, you may still have a lean situation at start, but I'd go back and redo the electrical connections to the starter and try again.
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