Of course your app with one driver would get usage. Your comment contains its internal contradiction right there in plain sight.

If "no one would use the app due to high wait times" then your driver would always be available and ready to go. So someone would be happy to use it whenever the dominant app is having trouble keeping up with demand. Which will always happen from time to time as long as demand has occasional peak periods and supply has some practical constraints, which a fleet of cars always will.

But drivers aren't locked in to a single service, and there's little to no cost for drivers to be on both Uber and MyStartup, so the network effect is diminshed.

You should read more into this, but even though switching seems easy in theory, the truth is different. Going through another driver application process / training takes time, and even then using two apps creates extra annoyance. You would be surprised how lazy People are even if there was a way to grow their earnings through switching.

You're right that people often make lazy decisions, but every single Uber I've taken in the last year had a Lyft sticker as well and vice versa.

Which doesn't necessarily indicate that they would drive 50/50 for both. It might be closer to 80/20 easily.

Users already have to make arrangements with traditional car services for early AM airport runs because there is no Uber of Lyft supply in most areas (outside of NYC, SF, LA).