• integrate Gnome Tweaks into the settings, maybe behind a "customize" button.

I've noticed they've done that with a few of the settings, but overall I don't think there's much appetite to do this.

• add system tray support.

I remember Gnome wanting to make a cross-desktop standard for this (the currently most popular AppIndicator system is a bit of a mess for multiple reasons, including UX, security, and some Wayland-related quirks). It's went pretty much nowhere as it would require not only multiple DE's (and Freedesktop org's) involvement, as well as third party dev support. I hope it happens at some point though.

I don't even know how you're supposed to access programs running in the background currently.

It's in the top bar, if you click the section where the part with the quick settings. It's there as a drop-down. It's an extra click though.

• add a toggle to put all opened windows on separate workspaces. Navigating workspaces with touchpad three-finger-swipe is a joy, faster and more intuitive than Alt+Tab or via the overview.

That's a cool idea. I actually really like the sound of that. Let's get you on the Gnome team :P