Until you drop a chain with clipless shoes with clipless shoes on, so you can't even shoe brake very well like BMX riders do. I had that happen with MTB shoes on, and I completely wore down one of the lugs. I've also accidentally unclipped, before I switched to running my Time ATAC cleats reversed which I think is an old messenger trick.
And that is why I only ride my NJS frame on country roads, not in the city anymore :)
I'm a strong advocate for front brakes, but to me it seems damn near impossible to drop a chain on a fixed gear transmission. Some foreign object would need to be overlapping the chain plane somehow, which sounds like you're already crashing anyway. I think it's far more likely for the chain to outright break.
I've had it happen a few times, and on a few different bikes. Usually the cause is a chain that is too loose, and especially if the wheel is cocked to the left due to the drive side of the wheel being pulled forward from hard pedaling.
One chain tensioner on the drive side is a good preventative measure, I prefer both sides since it makes things easy.
And that is why I only ride my NJS frame on country roads, not in the city anymore :)