Thanks for the question! This is a deep one and is going to take a bit of history, so let's go for a little walk.
League, since the very beginning, has had two pieces of client software - the out-of-game client and the in-game client. For brevity, let's just refer to "the client" as the out-of-game one - the in-game one isn't what this AMA's about. Originally this was implemented on top of Adobe AIR, and went through a major rewrite a few years back, implemented on top of web technologies.
This split is not something that many games do. I struggle to think of an example really - the vast majority of games, including the other ones Riot makes, implement their out-of-game experience in the same piece of software that the in-game experience is delivered on. This has a lot of benefits, but the biggest of them is that the same experts that are tasked with making the in-game experience snappy and responsive can apply the same techniques to the out-of-game experience. The artists know how to make optimized assets for out-of-game because they have to for in-game, designers....ect you get the idea.
I can't speak to why the original split happened in the long, long ago, but when it came time to re-write from the AIR client, League's in-game UI technology was in absolutely no place to hold all the features required to execute the out-of-game experience League deserves, so the split had to remain. Web technology was chosen as the new fundamental base because the in-game UI tech could not be brought up to capability in a time frame that matched the urgency of the project. League invested in all that it took to make that shift, and the result really is considerably better than what we had on AIR, lest anyone get rose-tinted glasses about that.
So back to the point - pros and cons of making it from scratch? Well, if we were to do so without changing anything fundamental, there's no reason to expect the result to be any better. In fact, it'd probably be a lot worse - the current implementation has gone through a lot of battle-hardening and while it has its problems, they're a lot fewer in number and lower in severity than a fresh implementation off the press would have. On the other hand, changing fundamentals can take a very long time, which makes the investment quite large.
When it comes down to it, there's two routes available - change the fundamental technology again, or iterate on the existing product. The ingame UI technology has advanced considerably since the last time it was evaluated for this purpose, which makes it the clear choice for the next big leap. In fact, the out-of-game experience of TFT on mobile is implemented on top of it. There's still a sizable gap to close there in order to capture all that League would need, and even if it were ready today there's a helluva lot of software to create if we were to bite off that project. For now, we're committed to iterating and improving the existing, but that does not lock out the option for us to undertake the huge project of moving to in-game tech somewhere down the road.