Search Engine is a great podcast and that was a great interview. For me it didn't really answer the question though, but I guess the answer is very individual.
Likely a lot of time was spent iterating and experimenting with different ideas, testing out concepts, tweaking, etc. Haven't played the game but I do work as a software developer.
Carmack actually developed Doom on NeXTSTEP, not Linux. The windows and DOS ports were not released as open source, due to copyright issues concerning the sound library. So they released the source of the Linux port instead. I don't think it made much of a difference in practice since it was a fairly popular game even before that.
Another vote for Debian stable with backports and flatpaks. I don't really have an issue with outdated software, and I really like "apt", maybe because I'm so used to it as this point. I've been running mainly Debian for 12+ years now.
My second choice for personal use would be Arch Linux. I had very good experience with it back in the day and their wiki is fantastic. But I'm too comfortable with the simplicity and stability of Debian at this point.
At work I use Ubuntu because everyone else uses it. It's not too bad. I just ignore all the crap I don't like (like snaps).
Looks like there is a recent port on steam.