Flash was one of those things that tried to do too much, and some of its things started being at cross-purposes with each other. The video conflicted with its roots as a vector/animation studio, and that's why Apple famously didn't use it - it ended up being a battery hog.
A lot of interests in web-based video wanted DRM, which meant it was never going to be usable by Free Software.
It was trying to do too much and then the usual corporate mismanagement led to its demise.
Apple famously didn't use it because it didn't sit inside their walled garden and they couldn't put any ads in it, take a cut of every sale, nor charge developers annually for the privilege of building on their property. That's 3 revenue sources that users and developers were skipping out on.
Android phones had Flash support and it worked well enough. Was it desktop-level, of course not. But Android native apps themselves were a mess from a power consumption standpoint until they implemented a JIT compiler in v2.3 "Gingerbread" as well as a task manager that freed up RAM by closing inactive applications. I remember specifically choosing an ASUS android tablet over an iPad because I wanted to use the full web, open multiple tabs at a time and play Flash games, rather than deal with iOS' "one app open at a time" philosophy.
Apple famously included a web browser on the iPhone, which puts a hole in your theory.
The Internet of 2007 was more than Flash, so of course it had a web browser. It was designed to compete with Nokia, Blackberry and Windows Mobile, all of which had web browsers. The App Store didn't even exist before the second iPhone model came out in 2008.