I have moved from Raspian to Hassio and Yes it’s still working. I have both local file and tts generated mp3 working.
I had mine on /config/www/audio folder. Folder permission 755
Make sure you can reach it from http://yourhaip:8123/local/audio/youraudiofilename.mp3 or what ever audio file extension you use.
https://yourdomain/local/audio/youraudiofilename.mp3 - make sure you forward 443 to 8123 on your firewall…
For tts cache folder
I have it in /config/tts didn’t make a chabge on the folder permission.
I tried /tmp/tts but the tts mp3 does not get written.