Most of the ubers and lyfts I've been in had both apps running. If a third app comes in with decent driver incentives, and can convince google maps to show them, they'll be able to get passengers and drivers. And, as you said, they only need to do it city by city.
I will say, when i needed a ride in a more rural area near Seattle, I did go right to Uber, and that driver wasn't on Lyft (but was considering it), so there is some stickiness, but I don't think it's much.
I see your point, but there has been a zillion attempts to do the Lyft/Uber thing and tons of money has already tried. The pie is just to big to ignore. So if hasn't happened already, it's not likely going to happen.
So yes, maybe a 'serious competitor' could try in some city, and as you say maybe 'word would travel fast' and maybe it might be possible to get a critical mass of drivers ... but then Uber/Lyft will just undercut them in a war of attrition.
To your mini-case: maybe there was a few other 3rd parties in existence, but you possibly never even heard of them. And even if the drivers sign up ... without the demand well there's no game anyhow.
It's been commoditized, so the advantage now is scale, brand etc. which the incumbents have, and they have money.
That market is done for now, until something comes from a different angle: new tech, new regulation, public transport shift, self driving cars etc.
You can't fight a subsidy war with uber/lyft now while they're still in growth mode. Once investors start demanding profits, and the prices go up, that's the time to come in and spend your money driving people around. Especially if you hear about others doing the same in other cities, or when some of the big loans are coming due.
It's a commodity, brand doesn't matter. Scale across cities doesn't really help that much. Money helps, but public market investors don't like to see it being spent quarter after quarter.
I will say, when i needed a ride in a more rural area near Seattle, I did go right to Uber, and that driver wasn't on Lyft (but was considering it), so there is some stickiness, but I don't think it's much.