Welcome, I'm Chris (aka Moose). Experts aren't employees / agents of Just Answer. We are independent service providers using the Site to sell their Expert knowledge to Customers. I strive to provide EXCELLENT 5 STAR service, let me know what's needed from me so your pleased.
I do not agree with changing the one bad coil is the fix here. The only thing that changes from starting immediate to waiting five seconds with the key on. Is the fuel pump has longer to run prior to cranking allowing it to build up more pressure. I notice you stated the pump was replaced already but I feel like its worth checking out.
Can you monitor the fuel pressure with a gauge in line on the fuel rail. You need to hit 62 psi minimum prior to cranking. If the pressure is slow to build you can have a restricted filter, weak pump, dirty pump relay contacts to blame.
The coil does need to be addressed but it will just cause a misfire, not this issue.
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