A ton of time and resources have been put into making web apps work across multiple rendering engines. These days, it can be difficult to have something break on one platform if you are using a modern development stack. The webview engines that Tauri uses are all quite modern, see https://github.com/tauri-apps/wry/discussions/202 for more details.
Assuming you meant Deno, we used Node for the CLI because most web devs already have npm installed. We want the getting started process to be as simple as possible. Our Node CLI is just a wrapper really, as nearly all of the commands are actually implemented in Rust. You don't even need Node to build a Tauri app anymore. Node is not including in a shipped Tauri app, although we plan on supporting both Node and Deno as optional replacements for Rust in the future.
If you scroll down to the roadmap, you can see that system trays are supported. We don't yet support native context menus, but that would be a relatively simple addition. Our GitHub readme has a table comparing Electron's and Tauri's features, do you think it would be useful to include that on the website landing page?
Having the getting started button at the end is a good idea I think, we'll look into that.
We say "web frontend" because that's as descriptive as we can be really. We support any web framework that runs in a browser. As for screenshots of big companies' apps, we don't have any major companies that have adopted Tauri yet, as we just came out of alpha, but we will add a section like that as soon as we can.
It's difficult for us to demonstrate how Tauri is used in short snippets that could fit at the beginning of the Getting Started section. Electron's site doesn't have any code snippets either, just snippets for installing via npm and some example apps.
The demo app has very basic styling, it isn't designed to be pretty. Since a Webview with HTML/CSS/JS is used for the GUI, you can style your app however you want.
We didn't take over control, rather we helped setup an independent org around webview and other related repos. At the time, the original author of webview expressed plans to work on it a lot. However, this didn't really happen. Webview is stuck with some nasty bugs and missing features, and none of the members of the Tauri team had enough C experience to fix it efficiently. Instead, we created our own pure Rust solution (https://github.com/tauri-apps/wry). We've already given it way more features than the original webview project, and it doesn't carry the bugs that plagued webview either. The next release of Tauri (about a month or so out) will use Wry, and will have features such as multi-window and fancy window styling (frameless, fullscreen, custom controls, etc...).
Servo is quite large and extremely difficult to build from source. We've tried to make it work with Tauri but decided it's not worth it, at least for now.