What happens when the web wants to implement some special optimization or security addition but SQLite doesn't want or support being used that way? What happens when SQLite makes a breaking change in a future version that might make sense for them, but would break a large number of web apps?
Once something is in the web standards, it's VERY hard to drop support for it. SQLite might make sense now, but what about in a decade?
No matter how you slice it, WebSQL has the ability to become a massive maintence liability, one that all major browsers agreed was too significant to codify into a standard.
You can replace every mention of "SQLite" in your comment with Chrome (or Google) and suddenly you're agreeing with the comment instead of expressing your opinion against it. Chrome currently dominates web standards, and other browsers must keep up or risk losing even more market share to Chrome.
Many web standardization systems already require multiple implementations of a feature before it can become "standard", and by that nature some will have to implement it first. Chrome has Google behind it, and they have a lot of money and a good reason to spend it here (the better the web is as a platform, the better their company which is based on the web works).
But there are plenty of examples of where Chrome's "dominance" didn't allow them to just unilaterally push through APIs or platform changes. PNaCl failed and we got WebAssembly which is much better. The original version of ShadowDOM was shot down and it left Chrome behind other browsers when it came to the true implementation. HTML Imports were again something chrome implemented first and is currently being removed due to it ultimately not working out well and the platform as a whole moving in a different direction. Hell most of the WebComponents spec was changed or removed before it started getting in other browsers in a major way.
Sure, Chrome could just stop improvements altogether, but that seems like a poor solution. One thing I'd love to see from Google is for them to pull their (in)famous move of eating their own lunch. I'd love to see them pull a Servo and try to spend the time and money creating a new rendering engine from scratch to compete with their own Blink engine.
Edge and Opera have only a fraction of the Firefox, IE, and Safari usage, so I'm not entirely sure why you're phrasing things this way.
I agree with your above comment, and we can argue that working on a single web-engine might lead to less duplicated work for Front-End devs, and less bugs in the web overall.
Given that sqlite is the most widely deployed DB already, making it part of web browsers doesn't seem like a huge deal to me. https://www.sqlite.org/mostdeployed.html