You don't have to choose between embedding the build time or commit information, but you do have to choose between embedding the build time or having reproducible builds. I know what I prefer, as a maintainer and consumer.
It's really nerve-wrecking (to me) when you pull down a binary from the website releases, pass it to sha256sum and gets something different than when you built the same commit from source. Usually that's because dates and times are embedded in the build, but sometimes stuff is just different depending on who built it, and that's scary.
To be fair, I'm aware that a poorly archived/copied file can lose them (especially over a network) and it's not part of `POSIX.1`. At the end of the day though I've found it's a reasonable "belt" to wear with your suspenders.
I needed this in pathological cases in the first place; think "this stuff dates from when this mature company was just a startup" kinda things. Knowing when it was built wouldn't have changed anything about the executable itself but would have helped with "which strata of old employees should I be hitting up", what other bugs were present, whether we were using a broken compiler at the time, etc.
In a perfect world, sure, none of that would matter, but.... the sibling reply from mentat is right in principle, but in practice sometimes the pathological does happen.
However, if I had to choose between knowing the exact commit and knowing the exact time, I'd certainly choose exact commit, no question. But you don't; you might as well just stick all the data in.