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No, not really. The cam(s) rotate at 1/2 the speed of the crank. A complete engine cycle is 720 crank degrees or 360 cam degrees. The crank pattern will therefore repeat twice for each engine cycle and the ECU needs to see the cam pattern to discern whether the engine is on the 0-360 or 360-720 phase. If it doesn't know this then it might be shooting fuel at a closed intake valve which is not very good for economy and emissions.
The majority of OEMs will not fire injectors until it gets this sync. Spark doesn't matter because the plugs won't do anything on the exhaust stroke because there isn't much left to burn in there. This also explains the observed behaviour and incorrect assumption that the injection timing comes directly from the cam. My subaru for example uses the cam sensor to start. After that you can unplug the cam sensor and it will happily run all day long. Won't start unless it gets that signal.
-Michael
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