Switzerland, where I reside, is similar in many ways (though at a somewhat smaller scale): the Zurich metro area has about 1.1 million people. Geneva and Basel metro areas are each around 500,000 people. The Bern metro area is about 350,000 people and yet the small suburb where I live (pop ~30,000 people) has a relatively large amount of competition: Swisscom (20 Mbps max) and Sunrise (30 Mbps max) each have DSL offerings, UPC Cablecom offers fiber-to-the-node with a EuroDOCSIS 3.0 coax last mile (currently the top plan is 150 Mbps max but this can increase in the future up to 400 Mbps), the electric company is running fiber to every property (it's at most homes now, with 90% availability in 5 years and 100% availability by 2020) and there's a variety of private companies that offer service over the municipal fiber. There's also several 3G and 4G mobile phone providers who offer service with varying speeds (up to 42 Mbps) and bandwidth caps with essentially total coverage.
In short: even with a relatively low-density city composed mostly of private homes and low-rise (under 4 floors) apartment buildings it's economically viable to have many competing firms providing high-speed connectivity. There's really no excuse why US cities like Houston, Phoenix, etc. shouldn't have a good amount of competition in regards to connectivity.
If anything, I'd posit that super dense cities like NYC and the like would be more difficult to run high-speed connections particularly due to the huge amount of legacy lines and equipment (e.g. gobs of twisted copper pairs in cable ducts where a modern fiber line would use much less space but replacing the copper would be disruptive and expensive) and the inability to just plop down equipment as needed due to limited aboveground space. In a lower-density city there's probably more room in cable ducts, places to put above or below-ground equipment boxes, less legacy cruft, etc. that should make it easier to build out high-speed networks and provide competition to customers. Ideally, things could be simplified by having a municipal fiber network that's owned and managed by the city (or, if they must, a contractor) but has service provided by competing private companies over that fiber.