Actually we all were dead already from birth to our natural mother. Adam and Eve died when they disobeyed God. Thus all of their offspring were born dead. Dead people cannot bear live children. Jesus confirms what people are without what he brought:
"And another of his disciples said unto him, Lord, suffer me first to go and bury my father.
But Jesus said unto him, Follow me; and let the dead bury their dead." Matt 8:21-22
But then this is why Jesus came to men:
"The thief cometh not, but for to steal, and to kill, and to destroy: I am come that they might have life, and that they might have it more abundantly." John 10:10
Would not then a second death be that which a born again person dies when after receiving the Life which Jesus is then choose to cast it away?
When they return to death that would be the 2nd time for them.
Actually, really, your explanation is based on what? There is a time for all unrighteous to die the second death, and also to be judged righteous to Gain eternal life.
Matthew 10:26-33: - Have No Fear
26 "So have no fear of them, I for nothing is covered that will not be revealed or hidden that will not be known. 27 What I tell you in the dark, say in the light, and what you hear whispered, proclaim on J the housetops. 28 And K do not fear those who kill the body but cannot kill the soul. Rather fear him L who can destroy both soul and body in hell. * 29 Are not two sparrows sold for a penny? * And not one of them will fall to the ground apart from your Father. 30 But M even the hairs of your head are all numbered. 31 Fear not, therefore; you are of more value than many sparrows. 32 O So everyone who acknowledges me before men, I also will acknowledge before my Father who is in heaven, 33 but P whoever denies me before men, Q I also will deny before my Father who is in heaven.
Notes: -
I Mark 4:22; Luke 8:17; [1 Tim 5:25]; For ver. 26-33, see Luke 12:2-9
J See Luke 5:19
K Isa 8:12,13; 51:12,13; Jer 1:8; 1 Peter 3:14
L James 4:12
* Greek Gehenna
* Greek assarion, Roman copper coin (Latin quadrans) worth about 1/16 of a denarius (which was a day's wage for a laborer)
M See 1 Sam 14:45
N ch. 6:26; 12:12
O [Rom 10:9,10; Heb 10:35; Rev 3:5]
P 2 Tim 2:12; 2 Peter 2:1; 1 John 2:23; [Mark 8:38]
Q ch. 7:23; 25:12; Luke 13:25
The killing of the soul is to be much feared as this is the second death.
However, even those who are candidates for the second death can be redeemed by God if they repent before Him their sins and will live.
Since the fall, all who have been born are candidates for the second death, but if we turn to God and repent of our individual sins against God then we will receive the gift of life by living by faith believing His promises to us.
Shalom
PS: - Consider carefully what you meant when you said that we were born dead but physically had a living soul.