Definitely not. The google product is not search or gmail or any of that stuff. The google product is dirt cheap computing. Everything else google does is just an attempt to exploit that competitive advantage. Running it on anyone else's stuff would erase the advantage.
Even in the collocation facilities google was developing this skill for example by installing their infamous corkboard servers with the drives attached by Velcro.
I meant the question more in the sense of starting off with AWS and moving to their own servers like it is popular for most startups to do nowadays. Though you make a good point about gaining the skills for a robust infrastructure early on being important.
It rarely makes economical sense to start with AWS. Very few people have bursty enough traffic for it. Most sites I've dealt with don't even have a day/night cycle that's pronounced enough to justify spinning up servers just for the daily peak until they're huge.
A lot do have a delusional expectation that building on AWS to handle that expected 10x overnight growth that they dream of justifies it, but that "just" justifies good caching and the ability to spin up frontends on AWS or similar if neeeded. Which ironically makes you likely to spend less on your dedicated hosting, as if your setup is ready to spin up cloud servers when the load goes too high, you can afford to get much closer to the wire before you add more dedicated hardware for your base load.
For most people, renting a dedicated server (there's a huge number of providers that charge month to month, with no commitment) will come out far cheaper for anything they use more than ~8 hours a day on average.
To me, if a startup puts everything on AWS, it's a sign they have poor cost control.
From experience at RBI I had one tool that ran on aws in hindsight it woudl have been cheaper to go with linode and that was a single small instance.
I noticed at silcon mikabout that prity much all the start ups where using aws - I suspect that 90% of them would struggle if they had to set up their own colo.
Even in the collocation facilities google was developing this skill for example by installing their infamous corkboard servers with the drives attached by Velcro.