Two brake positions on fixed gear/single speed
You can use cross top levers without the main lever with some hacking. There is a thread about it here: https://www.bikeforums.net/singlespeed-fixed-gear/108969-how-install-inline-brake-lever-primary.html

You can also replace it with a 'normal' brake lever. If your bar is oversize (31.8 mm) it can be hard to find a brake lever that fits, but it's certainly doable.

Having two levers might not be such a bad idea though, since it means you have a brake lever within reach from all hand positions. But I'm planning on installing just a 'top' brake lever on my fixie, so I can certainly see your point. You can also put a TT bar end lever at the bottom of the drop and use the cross top lever as well. That will look 'cleaner' than the usual road hood thing.