Hah, I have a different experience: my Teamspeak server literally stopped working once they started requiring a license key even for self-hosted instances. I scrapped the server entirely and switched to Mumble.
Ah, so your circle of friends does not include people with messed up audio. Teamspeak happily passes on whatever it got as input. Discord does quite a bit of filtering, able to salvage legibility of some of my friends.
It isn't a matter of server-side vs client-side (I have no clue what Discord does, but it does eat quite a bit of compute client-side), but of what we get out of the box.
Teamspeak never stopped working so there was never a reason to switch to anything else.