Why does linux not support setting timestamps for files?

I am legitimately confused how to go about doing this properly.

I want to run a command that will save ALL timestamps of files—birth, accessed, modified.

I know that "ctime" has nothing to do with the file creation but rather inode 'c'hange. While the technical side is interesting, it neither seems to solve my problem, nor provide me a way to do it (not that I could find).

I need to back up data best I can, and I want a tool that can do it properly. Is it possible at all on Linux?

I used rsync but that does not work.

$ rsync --times --crtimes --atimes --open-noatime ... rsync: This rsync does not support --crtimes (-N) $ rsync --version rsync version 3.3.0 protocol version 31 ...

I went and looked around, and it seems that there is no syscall that exposes this feature.

I am confused. Why does linux not support setting timestamps?