Need a bit more information. What's the mileage on the car? Maintenance history - what has already been done, tune-up / repairs?
What exactly do you mean by restricting airflow? Blocking the intake by your hand while someone cranks the car or similar.
One thing that strikes me off the bat - by restricting the airflow to the engine to help it start - points at a possible issues with the cold start injector, the throttle body (could just need a good cleaning), or a potential fuel pressure / fuel supply issue on cold starts.
By restricting airflow as you crank the engine, you effectively "choked" it down - enriching the mixture to help it ignite readily. Same effect as older carburated engines - they had a choke that restricted airflow to enrich the air/fuel mixture as well help "pull" fuel into the cylinders.
Excessively worn sparkplugs and/or worn ignition components (plug wires, cap, rotor, etc.) can cause a hard start issue as well. If they are not up to spec, i.e., spark is not "hot" enough or even a misfire issue, they would have a very hard time igniting the air/fuel mix on cold starts.
Dirty and clogged injectors are another possibility - by choking the induction system - you could have temporarily "flushed" dirt and grit out of the injectors when there was a stronger vacuum present. This could also point at other potential culprits - like: a dirty fuel filter, dirty or worn injectors, fuel leak (bad seal in injector log, hardlines, etc.), bad fuel pump, bad pressure regulator, etc.
2002 Corolla S, Silverstream, 1.8L 1ZZ-FE (VVT-i)
2003 Matrix XRS, Cosmic Blue, 1.8L 2ZZ-GE (VVTL-i)
