Great attitude.
In an Olympic race, all run in the race, but only one receives a gold medal. That does not mean that those who do not receive a gold medal are disqualified from the Olympics. So runners in the Olympics may receive a corruptible crown, but believers may receive an incorruptible crown and the Bible mentions 5 different crowns. The crown of righteousness, the crown of rejoicing, the crown of glory, the crown of life and the incorruptible crown. (On the head of Jesus were said to be "many crowns" in Revelation 19:12.
So "run to obtain" but out of ALL who run ONLY ONE receives the prize. Do we work for and earn the free gift of eternal life or the prize?
Who said anything about a reward system for reprobates? Those who are disqualified for the prize do not receive the prize, which is not a free gift.
Of course he did as all genuine believers keep the faith. Paul does not seem to indicate any insecurity about his position - Finally, there is laid up for me the crown of righteousness, which the Lord, the righteous Judge, will give to me on that Day, and not to me only but also to all who have loved His appearing. (2 Timothy 4:8)
It's actually you who is absolutely wrong with Paul's concerns about losing salvation. We don't even find the specific words "lose or lost salvation" in the Bible.
Other translations say disqualified/disqualified for the prize.
Still doesn't mean a loss of salvation in 1 Corinthians 9:27.
For a deeper study see -
https://www.preceptaustin.org/1_corinthians_927_commentary