I have a controversial opinion.
We got to the canonical "LAMP" stack incrementally, shaped by the computing environment at the time. The servers that you had available to your web app 15 years ago were maybe 2 cores running at 1.5GHz and you had a couple gigs of RAM, and you wrote the app in a very slow dynamic language. So in order to scale out to the Internet, you had to split your application into replicas that ran on different servers; the traffic you were getting and the architecture of your application meant that you needed more than 1 CPU second per second, and that was the only way to get that at the time. All of these replicas needed some coordination, so you had a database server to handle that coordination, and it ran on the biggest computer possible because it was the single point of failure. (And when one computer wasn't big enough, it was time for ugly hacks like sharding.)
Fast forward 15 years and the landscape has changed dramatically. You can get 448vCPU machines from AWS. You're writing your application that is compiled or JIT'd into native code. With the same hardware, your application of today might be 10-100x faster than your application 15 years ago on the same hardware. But the hardware is even faster, and computers are bigger (you can go get a 64 core processor off the shelf at an electronics store!). So one node might now be 1000-10000x more powerful than where this "I must have a hundred replicas of my application" mindset came from.
With that in mind, if you start right now, you can build a web application that scales beyond your wildest dreams with a single computer. If you only have one computer, why not build the database into the application? You will have made so much money by the time the 448 core computer is inadequate for your application that it doesn't even matter. Hire 100 developers to move the thing to FoundationDB or Spanner or Postgres or whatever "real" database you think you need to scale to the next 10 billion users. (Interestingly, the top article on HN right now is "What if it changes?", a sarcastic reminder that maybe today you should build the system you need for today. If it sucks in a year, fix it in a year.)
I am throwing some pretty important things out the window. You probably update your application, so you want some rolling deployment that aborts if the new version fails to start up or whatever. A tornado blowing up your datacenter on your biggest use day of the year would be bad. Your users are all over the world, so you probably want to serve as much content as you can from a computer near them. But honestly, the state of the art for these concepts are pretty new. Globally consistent ("planet scale") transactions are hard, and not that fast. Automated canarying is also not trivial. So you might build a really complicated application to support those needs, and not even achieve them, because nobody has achieved perfection there yet.
Anyway, my controversial opinion is: don't build a distributed system unless you are absolutely sure you must have a distributed system. Every time you split up your state storage, your work as a programmer becomes harder. A single thread; no need for transactions, everything happens in order. A single computer with multiple threads; you'll need some locks, write barriers, or atomic instructions. Multiple computers with multiple threads? Now you're writing a computer science paper. That's a super fun activity, but check that discovering new classes of computer science problems is what your company's business is before you go out of your way to start doing it. You might be able to make a lot of money by being pretty boring. When you're relaxing on your private island, you might find a recreational activity even more fun than finding bugs in distributed transaction protocols. Who knows.
The currently-accepted simplification is to use a single database server to coordinate your stateless application replicas. That doesn't protect you from tornadoes, give users in Antarctica sub-millisecond page loads, or let you upgrade the database without downtime. If you're OK with that, it's totally reasonable to just build the database into your application as SQLite does. It's simpler, and no worse to an outside observer.
(That said, I tend to reach for Postgres first because it has a lot of polish that I don't think SQLite has. But I don't think anyone is unreasonable or stupid for picking SQLite. Especially if the SQLite instance streams its data to S3 and you have point-in-time recovery options for a disaster or bad release. You're going to want those with Postgres too, and they aren't enabled out of the box.)