I'm a bit late to the party, but given that the thread still gets bumped once a month or so, I guess I'll chip in. There's a lot of posts that misunderstand Qt and why QtWebEngine is such a hog - so as a programmer who worked with Qt, I'll clear it up a bit.
EA uses the Qt framework for its GUI client. It's cross-platform, pretty easy to use, and in general not a resource hog by itself. Quite a few programs use it, and they mostly limit themselves to basic Qt functionality, which makes them pretty lightweight.
While Qt offers a lot of functionality, it's not a web browser - it only uses a small subset of HTML in some places, and it does not have a Javascript interpreter. Of course, sometimes you have a ready-made webpage with all the GUI bells and whistles and JS in place, and you'd just like to transplant it to the desktop client.
There are actually a lot of solutions for this - there's Electron which is basically a repackaged Google Chrome, used by e.g. Discord (and at the same time looked down on some for resource hogging 🙂 ), Valve has a web browser within Steam as well (you might've seen the steamwebapphelper.exe process). Qt also offers a solution, but things get slightly complicated here.
In Qt4, the web browser module was called QtWebkit, and used the WebKit engine underneath. For some reason, in Qt5 this was replaced by QtWebEngine - maybe it was less maintenance, or maybe WebKit did not have the functionality, I don't know the details. Anyway, QtWebEngine uses repackaged Google Chrome underneath instead, and QtWebkit is deprecated right now. The process you're seeing - QtWebEngineProcess.exe - is basically a reskinned and renamed Google Chrome process.
Since the process is Chrome, there's probably a myriad reasons why it's a resource hog - maybe chrome devs configured it to use a lot of resources by default, just like a standalone web browser. Maybe Qt developers didn't catch these options, or expose the interface to fine-tune them. Maybe Origin GUI client devs didn't have the resources to fine-tune the browser and were only able to replace QtWebkit before it became deprecated. Either way - the fact that Origin was released in that state either means that someone neglected testing, or that management set (or was forced to set) deadlines so short that there wasn't time for anything else. That's a show of bad management, on some level.
Now, comparing to Valve - apparently Steam uses something called VGUI, their own GUI toolkit they started way back for half-life 1. Back then there weren't GUI toolkits as fancy as Qt, so Valve rolled out their own. It apparently uses Webkit for web browsing, and since Valve developed the entire GUI toolkit, they can fine-tune it for Steam's purposes. Of course, that took a lot of resources to do, more than a small crew of say, 4 devs, required to make a Qt client. There are ways to make QtWebEngine more resource friendly, I assume - starting by profiling what EA webpage contents are causing the hogging and fixing them, compiling slightly customized QtWebEngine to limit resource usage, maybe even hiring devs to work on (open source!) Qt and Chromium to squeeze out more performance. To what extent EA is willing to hire extra developers for all that remains to be seen.