Let's come at the question a little differently. What purpose does a key signature serve?
Generally speaking -- very generally and simply speaking -- as humans we expect songs to have movement, expressed via tension and resolution. Even people who are musically uninclined or self-describe as 'tone-deaf' are typically left hanging when one plays a scale from the tonic to the leading tone and then refuses to play the octave (do re mi fa so la ti..............???)
If I establish a certain tonality or key around whatever root note I choose for the day, and I play a V7 chord, most listeners will expect the song to eventually (and relatively quickly) come back to the I chord. The full cadence V7 -> I is the driving force behind a lot of Western music - so much so, that it compels players to use it to step outside the key for the sake of that movement to show up for other chord changes.
Consider a 12-bar blues that makes use of a series of b7 chords. If we adhere to Mrs. Oldface's music theory class from high school, upon seeing a dominant-seventh chord this should be a giant flag that one is in a key that's a perfect fourth higher than the root of the chord (or ah, a perfect fifth down, I guess, depending on how you're feeling). But when we're in C, for example, and we use the C7 to 'predict' the IV -- F -- only to return to the I and then move to the real V, being G -- all of a sudden we're off in nonsense land as far as Granny Oldface is concerned. That dominant-seventh interval has no business in a half cadence -- but of course, a C7 -> F progression is totally normal, if we're in F major. In other words, if sticking to the idea that one key should rule them all, the only time the music theory sticklers of the world will 'let you get away with that' is when you're moving from the V to the I.
So, did the song modulate to F for 4 bars? Is that a useful approach to playing, arranging, or improvising over the 12-bar blues example? Or instead, did the composer/band/whomever simply capitalize on the notion that any chord with a tritone in it can be used to 'pull' the listener's ear from where the song is now, to where the song is going?
Now tritones invite a lot of tangential discussions about other ideas like chord substitution and that (the tension of the B-F interval will resolve to C as easily as it does to F#) and I don't want to just range all over the place. But if we're wondering why one key is no better or worse than all keys, consider for a second that key signatures are as easily discarded as they are adhered to, without inducing cacophony in the process.
This doesn't explain why every song isn't in C, since this could all just be happening in C and we could just be telling people to shove off whenever we play a Bb in a song relative to C major. But it illustrates in a roundabout way that implying a strict tonality of "X Major" and its relative minor is practically a rule that's made to be broken, precisely because that V7 -> I tension and resolution is a powerful method of informing the listener to your next move and as such, to only apply it to G7 -> C for all of eternity would be unfortunate.
So it isn't terribly hard to imagine, even if it is just somewhat hyperbole at this point, that in an imaginary world of 12 tones but only 1 key, that a player somewhere might realize the potency of that tritone resolution and say to themselves, "Hunh. I can use a I7 to signal a move to the IV chord, I wonder... I wonder where else I can go..."