If you use the translation, "sin, when it has run its course", you could liken it to a person running a race, determined to get to the finishing tape. That would be to complete the course. It matters not here if there are millions, or billions of humans all seen to be running this race. Every one of them will complete the course because every one of them will die. "For the wages of sin is death" we are told elsewhere in the New Testament (Romans 6:23a), which agrees with what James said, though using a different way of illustrating the point.
All humans have the sinful nature that guarantees they are on that course, being driven by sin, which leads to death. Whether they run, or walk, or crawl, they will one day die, proving that sin had (sooner or later) claimed them in death. There is, however, a wonderful, divine intervention that will mean eternal joy despite having to die physically. The rest of that sentence quoted above states what that is:
"...but the gift of God is eternal life through Jesus Christ our Lord." Romans 6:23b A.V. (See also Ephesians 4:2-9.)
Back to the one verse in question, James 1:15. The A.V. simply translates it as:
"Then when lust hath conceived, it bringeth forth sin: and sin, when
it is finished, bringeth forth death."
As all people are sinners, that is their default position before God. Yet the verse in Romans shows that God has established his triumph over sin and death, through what the Son of God did to defeat the power of sin, death and the devil. Now all who admit that they are sinners who need what Jesus did to liberate them from their slavery to sin, put faith in him (no longer in their own efforts). James says in verse 12 that the person of faith is able to endure temptation. And, as in verse 21, James says that having laid aside all filthiness (sin), those people of faith receive with meekness the engrafting of God's word, which is what saves their souls.
The final outcome for them is radically different to those who go along with their lusts and sinful desires. They will still have to receive their wages for the body of sinful works they did, and die physically (as per Romans 6:23a) but death will just be a blip (as it were) for they have already passed over from death to life the moment they believed (John 5:24). Death will not be able to 'hold' them because they have been freely pardoned as in Romans 6:23b. They found the narrow gate (Christ), entering the narrow path that leads to life eternal, not eternal condemnation as is the final result for all on the broad road paved with lusts, temptations and sin.
There just could not be any greater encouragement for those who have been liberated from sin's chains to resist temptations. Alas, those who are still captive to their lusts and desires are being pulled along to the finishing line by invisible chains. James starts chapter 1 saying faithful believers can have joy when tempted because their patience means they will not give in to temptation. They have wisdom from God to see the matter from above, from where God pours out on them every blessing and good gift (verse 17.)
Although the question is about James' teaching of sin in just one verse in his epistle, the whole chapter makes it clear, no matter whether the translation of verse 15 says of sin, 'run its course', or 'when it is full-grown', or 'gives birth to', or 'when it is finished'.