Is there anything cooler than a Fender Esquire or Gibson Junior? Their rough 'n ready, minimalist looks sure capture my heart. But there's more to it: these stripped-down instruments let you focus on your playing technique. There's no options, no second pickup to help you out when you need a different tone. It's all down to your hands to make music.
You'll really learn how to focus on your picking dynamics, on the difference fingers and picks (and pick materials make), and on the position where you pick - closer to the bridge for a sharper, twangier sound, or closer to the body for a mellower tone.
And then, there's the sound quality of the instrument itself. With less magnetic pull, notes ring longer and clearer. Your sound becomes punchier and broader - "more" to start with and mess up later on. Especially with the Esquire, which also takes the tone pot out of the equation in certain settings, you get a direct path from your pickup to your amp (heck, you could even remove the volume knob - I've done it). The end result is a fatter, punchier tone, and often more sustain as well, compared to a two-pickup model.
I know your volume and tone pot are the only non-human bits on your guitar that give you a direct way of shaping your sound, but I would really recommend trying to take at least the tone pot out of the circuit. Your sound should open up and gain a bit more shimmer. You can still take some treble out by rolling the volume knob back a notch. This works especially well if like to push your amp with your guitar itself instead of with an overdrive pedal.
Obviously, there's more than just the Esquire or the Junior. A lot of iconic Jazzboxes (the Gibson Howard Roberts Custom comes to mind) make do with a single neck pickup. A lot of "student" models came with one pickup - cheap, but probably also making them much nicer to play than if they had more.
So could you live with a guitar like this? Heck, yes! Sure, if you need to cover a lot of ground in different bands or a cover band, it might not give you the tonal variety needed - hey, at least you have an excuse to get a new guitar. As a signature instrument, however, it certainly works, and if the previous scenarios don't apply to you, I think you might be very happy with a single pickup guitar as your only instrument.