By the rules, if the entire party is rolling for stealth, only half of them are required to succeed. It's called a group check.
If your fighter was the only one with a bad roll, and your table doesn't use crit fail/crit success on skill checks (which is a homebrew rule, albeit fairly popular), then the result is probably legal.
I've never listened to any RPG podcasts but this is surprising given that I always see comments talking about how much it breaks the game. Do you think it's because they're making an entertainment broadcast rather than just playing a game, so they care more about crazy shenanigans rather than their individual experiences?
I'm a casual D&D fan since my only exposure is from Not Another D&D Podcast but I think it adds to the overall story telling experience. Super charges the lows and highs if it's a 1 or 20 especially on an important role. Does it break the game? Eh, not that I can tell and I've listened to hundreds of hours of the podcast. Though this is my opinion and not based on D&D rules, history, etc.
I also highly recommend Not Another D&D Podcast if you like silly shit mixed with crass humor, some good emotional content, and players fucking with their DM.
Maybe "break the game" is the wrong term, but the classic argument is that the strong fighter shouldn't be unable to lift a pebble 5% of the time or a player shouldn't be able to jump over the moon 5% of the time. Of course the counter-argument is that the DM shouldn't tell you to roll for things you can never pass or fail, but sometimes you need the illusion of higher stakes.