One of Uber's massive problems is lack of predictability. When going to/from the airport I can't fuck around with surge pricing, I need to be able to have a travel budget that doesn't change sharply for situations out of my control. I don't want to miss planes. A service like Blacklane (which by the way is now in the US and is growing very quickly yet seems to rarely make the articles that talk about Uber's competition) neatly solves that problem for you, and generally has better service and more professional drivers.
I am a frequent flier and your idea that there's zero-incentive to have a backup or alternate option to Uber ready to go doesn't resonate with me. Uber isn't reliable enough to forget about other options, and as long as those exist and are viable Uber's supposed "monopoly" strategy will likely be regarded in retrospect as expensive and stupid waste of capital.
The other alternative are exec cars which Uber offers but which will often be pre-booked locally.
I think Uber could get many business travellers on board if they'd offer some rewards system (similar to airlines). Otherwise there's no reason to be loyal to Uber if a local cab or some other hire service (e.g. Lyft, Addison Lee in London) are faster to use.
If your goal is to save time and mitigate risk, then pre-book with a local service before you even depart. Or, as many business travelers do, ask the company/person you're meeting to arrange a car to collect you. Then you just have to look out for one of those drivers in Arrivals holding a card with your name.
Many of the international airports I've traveled to have excellent transport links to the city centre, primarily by rail. They're quick, cost effective and low risk.
I don't see a compelling reason to book with Uber on arrival.
While I agree with your point, rarely are the rail links less risky to find/buy tickets/navigate than simply walking out the door at arrivals and catching a cab door to door.
The two airports I fly into quite regularly are London Heathrow and Vienna. At both locations I'd prefer to catch the Airport Express train than mess about with taxis in traffic.
I was in Hong Kong a few months ago and it was extremely straight forward to catch the airport train. Very, very obvious where to go, how to buy tickets etc.
Try catching a cab from Sydney Kingsford Smith airport into the city during peak hour. You will sit on the Eastern Distributor for a long time.
I'm not even a frequent flyer, but I'd estimate that a good 80-90% of my Uber usage is travel-related. At home, I have tons of other options to get around, but on a tight schedule in an unfamiliar city, I basically default to Uber all the time, because I rarely have the time or risk tolerance explore alternatives.
One of the things I love about Uber is that I can use it in SF, Singapore, Manila, Helsinki, wherever. It also works sufficiently well that I have near-zero incentive to try out Lyft/Grab/Ola/whatever, particularly since this usually involves yet another tedious sign-up process of entering credit cards and PIN confirmations and yadda yadda.