If I'm not completely mistaken, TS has a text chat too but not as good as Discord has. And it is on the same time as voice chat.
Not sure if Discord voice has a text chat at the same time too, I have not tried voice yet.
Btw, what is the "official" abbreviation for Discord?
You can be in one voice at a time, but you are technically in all text channels you have access to all at once on discord, though you can only view them one at a time. You can of course be in voice and text at the same time. Voice also works from a web browser.
I believe the only things the app does over running it in a browser is enable the voice overlay, and maybe allow you to use a direct joystick button mapping for push to talk.
I am genuinely curious what problems anyone may be having with it, as it should be fairly turnkey.