This is exactly how it works. There is at least 1 unbroken chain of descent from the first life to you. To each of us.
It's only recently that new life can be created by other means. E.g. nowadays it is possible to remove the nuclear DNA from an egg and transfer it to another egg.
Well if you want to get technical there is a 1 in a trillion xN chance that something really weird happened and your ancestor got some DNA from a dead thing.
It's, perhaps, soon going to get much weirder than that.
How much gene editing (e.g. CRISPR[1]) before the resulting life is "artificial"?
A few edits to fix a genetic flaw such as cystic fibrosis or sickle cell anemia? What happens when it's possible to edit out 100 such genetic flaws?
And do we go beyond obvious flaws? What if we figure out how to give everyone the visual acuity of Chuck Yeager? How many more "improvements" before the result is something akin to Frankenstein's monster?
Biotechnology will probably be the biggest growth area of the 21st century.
It's only recently that new life can be created by other means. E.g. nowadays it is possible to remove the nuclear DNA from an egg and transfer it to another egg.