You already said that you would have felt like something was off. That's detracting from the experience even though you don't consciously know why.
A ton of effort goes into choosing the fonts of the hundred different things you see every day, so you've been trained to unconsciously associate different fonts with different feelings. You don't need to be able to pick a font out of a lineup in order for font choice to affect your experience. If I spend enough time on Mozilla's site, I'm going to start associating Zilla Slab with internet-related things, even though I still can't distinguish slab fonts. Your brain's good at things like that.
For those who don't notice consciously it'll do something on a subconscious level. For those who do notice the difference between Arial and Helvetica, it can make a world of difference.
So you do get it, just on a subconscious level! That's exactly what good font design is all about: being invisible. It just so happens that being unnoticeable is incredibly difficult to achieve, and changes all the time with culture. A good font matches the tone of your message while remaining clear and understandable. That's because fonts aren't for robots, they're meant to be read by humans.
What Mozilla is doing here is actually a branding exercise!